11524:
11236:
5174:
52:
4752:
5353:
3818:
3759:
3774:. In particular, most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists essentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (no reference, no link to articles about the methods of computation that are used, no sketch of proofs.
11519:{\displaystyle {\begin{aligned}\mathrm {U} (n)&\supset \mathrm {O} (n)\\\mathrm {USp} (2n)&\supset \mathrm {O} (n)\\\mathrm {G} _{2}&\supset \mathrm {O} (3)\\\mathrm {F} _{4}&\supset \mathrm {O} (9)\\\mathrm {E} _{6}&\supset \mathrm {O} (10)\\\mathrm {E} _{7}&\supset \mathrm {O} (12)\\\mathrm {E} _{8}&\supset \mathrm {O} (16)\end{aligned}}}
2079:
4836:
9109:
5974:
4438:
7990:
8755:
4184:
12078:
In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group.
1952:
6256:
has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted
5169:{\displaystyle {\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}}
8974:
4747:{\displaystyle {\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}}
5869:
3415:
9749:
9574:
6541:
The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.
1164:
8959:
3629:
2652:
8589:
7744:
10313:
8630:
9399:
3133:
7352:
4070:
7736:
7639:
6980:
1556:
vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by
11082:
7539:
2074:{\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}
4290:
6708:
7132:
9260:
8052:
6628:
9182:
8218:
1046:
9104:{\displaystyle {\begin{aligned}\mathbf {C} &\to \operatorname {SO} (2,\mathbf {R} )\\x&\mapsto {\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}},\end{aligned}}}
12296:(if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.
6844:
3518:
7191:
2329:
2162:
10210:) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the
5969:{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}
3731:
2912:
10175:
For the usual orthogonal group over the reals, it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
3321:
6803:
11165:
5427:, which consists of two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so
11009:
10809:
9580:
9405:
11212:
10089:. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension
11241:
8979:
8883:
8635:
8492:
4841:
4443:
12002:
8878:
8868:
10731:
3552:
2550:
10099:
In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
3238:
8487:
8809:
8422:
7985:{\displaystyle A_{1}A_{2}={\begin{bmatrix}a_{1}a_{2}+\omega b_{1}b_{2}&a_{1}b_{2}+b_{1}a_{2}\\\omega b_{1}a_{2}+\omega a_{1}b_{2}&\omega b_{1}b_{2}+a_{1}a_{1}\end{bmatrix}}.}
10854:
10536:
8750:{\displaystyle {\begin{aligned}\mathbf {T} &\to \operatorname {SO} ^{+}(2,\mathbf {F} _{q})\\x&\mapsto {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},\end{aligned}}}
10494:
1547:
10243:
6538:, that is if a change of basis transforms the matrix of the first form into the matrix of the second form. Two equivalent quadratic forms have clearly the same orthogonal group.
1687:
9271:
8364:
1359:
9948:
Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the
9840:
to the power of the
Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.
4060:, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the
497:
472:
435:
3057:
10669:
9940:
Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the
10883:
The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.
4179:{\displaystyle \operatorname {O} (0)\subset \operatorname {O} (1)\subset \operatorname {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\operatorname {O} (k)}
10192:, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part
7292:
12079:
Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any
7644:
7547:
6927:
972:. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.
12019:
In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.
11016:
7454:
2350:
identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its
10733:). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors
10404:
of the kernel of the spin covering. The cohomology is non-abelian so that this is as far as we can go, at least with the conventional definitions.
3836:
2402:
is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.
8471:
For finishing the proof, it suffices to verify that the group all orthogonal matrices is not abelian, and is the semidirect product of the group
4223:
6658:
799:
7072:
2806:
equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.
9952:
is 2. A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector
9187:
7998:
6559:
9117:
8124:
2362:
1441:, since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.
6808:
3442:
10225:). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of
7137:
2275:
2108:
3683:
3410:{\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},}
2827:
10919:
12236:
10610:
is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to
9744:{\displaystyle \left|\operatorname {O} ^{-}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right).}
9569:{\displaystyle \left|\operatorname {O} ^{+}(2n,q)\right|=2q^{n(n-1)}\left(q^{n}-1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right),}
6770:
11113:
2962:
357:
10960:
10746:
12612:
12443:
11171:
1159:{\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}
12292:
equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be
1192:
that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the
12154:
6155:, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.
307:
11889:
8954:{\displaystyle {\begin{aligned}\mathbf {R} /2\pi \mathbf {R} &\to C\\\theta &\mapsto e^{i\theta },\end{aligned}}}
950:
around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see
11697:
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. These subgroups are known as
12638:
6641:
5979:
which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing
1621:
and the group of translations. It follows that the study of the
Euclidean group is essentially reduced to the study of
1390:. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are
792:
302:
11963:
12656:
12578:
12496:
11725:
11561:
10331:; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The
8814:
6549:
asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form
5393:
3854:
3799:
3624:{\displaystyle \left(\varepsilon _{1},\ldots ,\varepsilon _{n}\right)\mapsto \varepsilon _{1}\cdots \varepsilon _{n}}
2647:{\displaystyle {\begin{bmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{bmatrix}}.}
1641:
of a
Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of
12749:
11717:
10674:
8454:
8307:
5610:
11640:, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar:
12600:
8584:{\displaystyle {\begin{aligned}\mathbf {Z} /(q+1)\mathbf {Z} &\to T\\k&\mapsto g^{(q-1)k},\end{aligned}}}
6546:
3194:
2399:
1733:
11957:
11729:
11552:, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between
10328:
2193:, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to
915:
718:
5189:
The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.
12744:
12699:
12604:
12159:
8763:
8380:
6745:
6090:
2406:
1565:
785:
10308:{\displaystyle 1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1}
12219:
10817:
10499:
8242:
is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of
6086:
10460:
1475:
12694:
12197:
12034:
11845:
11757:
6080:
5984:
2239:
1651:
402:
216:
10954:
compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:
9394:{\displaystyle \left|\operatorname {O} (2n+1,q)\right|=2q^{n^{2}}\prod _{i=1}^{n}\left(q^{2i}-1\right),}
6719:
4383:: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.
4057:
1745:
1211:
The name of "orthogonal group" originates from the following characterization of its elements. Given a
134:
12689:
3542:
12224:
11587:
8317:
5566:
5216:
4314:
1314:
6375:
has two connected components. The component of the identity consists of all matrices of determinant
6158:
The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted
3128:{\displaystyle {\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},}
12739:
12346:
11597:
9833:
9800:
in case the element is the product of an even number of reflections, and the value of 1 otherwise.
5664:
5371:
3777:
3241:
3048:
2258:
1850:
600:
334:
211:
99:
7244:
cannot be both zero (because of the first equation), the second equation implies the existence of
480:
455:
418:
12350:
11097:
10939:
10875:
of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.
10332:
5824:
4028:
3934:
2717:
1810:
1169:
10647:(over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form
10557:
10007:
8311:
7134:
because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix
4763:
3267:
2945:
has two irreducible components, that are distinguished by the sign of the determinant (that is
1553:
1212:
947:
750:
540:
10654:
7347:{\displaystyle {\begin{bmatrix}a&b\\\varepsilon \omega b&\varepsilon a\end{bmatrix}},}
11557:
11553:
10441:
5375:
4322:
3939:
3921:
3781:
2930:
2926:
1787:
878:
624:
17:
7279:. Reporting these values in the third equation, and using the first equation, one gets that
12666:
12622:
12279:
12172:
11739:
11706:
11627:
10230:
10133:
7731:{\displaystyle A_{2}={\begin{bmatrix}a_{2}&b_{2}\\\omega b_{2}&a_{2}\end{bmatrix}}}
7634:{\displaystyle A_{1}={\begin{bmatrix}a_{1}&b_{1}\\\omega b_{1}&a_{1}\end{bmatrix}}}
6850:
is a non-square scalar. It results that there is only one orthogonal group that is denoted
5980:
5679:
3915:
1283:
1029:
861:
853:
564:
552:
170:
104:
12674:
12588:
12506:
12453:
3832:
1770:
of the determinant, which is a group homomorphism whose image is the multiplicative group
8:
12734:
10635:
8424:
is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the
6170:. Moreover, as a quadratic form and its opposite have the same orthogonal group, one has
5785:
5363:
3769:
1910:
1642:
1395:
1181:
835:
139:
34:
3177:
2395:. This results immediately from the above canonical form and the case of dimension two.
1203:, since the condition of preserving a form can be expressed as an equality of matrices.
12261:
10872:
7382:
For further studying the orthogonal group, it is convenient to introduce a square root
6535:
5336:
3963:
3189:
2792:
2751:
1775:
1610:
1595:
1445:
1372:
124:
96:
12711:
10053:
of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since
3290:} by permuting factors. The elements of the Weyl group are represented by matrices in
12652:
12634:
12608:
12574:
12492:
12439:
12063:
11565:
10553:
10185:
10050:
6480:
6281:
4773:
4304:
4202:
3974:
3891:
3873:
2682:
2190:
1638:
1263:
886:
874:
529:
372:
266:
10814:
This description applies equally for the indefinite special orthogonal Lie algebras
3307:
factor is represented by block permutation matrices with 2-by-2 blocks, and a final
3169:, the maximal tori have the same form, bordered by a row and a column of zeros, and
1196:, or, equivalently, the quadratic form is the sum of the square of the coordinates.
695:
12670:
12584:
12562:
12502:
12449:
12207:
12046:
12028:
11786:
11721:
11710:
10951:
10226:
10146:
10079:
5624:
5457:
4387:
3737:
coordinate to make their determinants positive, and hence cannot be represented in
2665:
has a more complicated structure (in particular, it is no longer commutative). The
2182:
1767:
1573:
919:
680:
672:
664:
656:
648:
636:
516:
506:
348:
290:
165:
12438:. Graduate Texts in Mathematics. Vol. 251. London: Springer. pp. 69–75.
10196:, as far as the discovery of the phenomenon is concerned. The first point is that
10006:
is the quadratic form associated to the orthogonal geometry. Compare this to the
6992:
is even, there are only two orthogonal groups, depending whether the dimension of
6119:. In other words, there is a basis on which the matrix of the quadratic form is a
2475:) and all objects with spherical symmetry, if the origin is chosen at the center.
12662:
12618:
12488:
11535:
10923:
10234:
10189:
9844:
6975:{\displaystyle \textstyle {\begin{bmatrix}1&0\\0&\varphi \end{bmatrix}},}
6337:
6120:
5996:
5987:
for the homotopy group to be removed. The first few entries in the tower are the
5860:
5668:
5542:
5445:
4016:
4005:
3248:
2755:
2739:
1806:
1583:
1376:
1239:
1200:
923:
894:
843:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
4325:, which is an orthogonal group one dimension lower." Thus the natural inclusion
12570:
12354:
12241:
11798:
11794:
11773:
11077:{\displaystyle \mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)}
10614:
representations of the orthogonal groups, and representations corresponding to
10197:
10119:
9925:
8433:
8377:
one can reconstruct a corresponding orthogonal matrix. It follows that the map
7534:{\displaystyle xy=1,\qquad a={\frac {x+y}{2}}\qquad b={\frac {x-y}{2\alpha }}.}
7032:
6527:
6311:
6307:
5692:
5512:
4040:
3047:, for every maximal torus, there is a basis on which the torus consists of the
2934:
2681:
are strongly correlated, and this correlation is widely used for studying both
2537:
2510:
2479:
2450:
1302:
1173:
946:, generalizing the fact that in dimensions 2 and 3, its elements are the usual
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
12407:
6360:, complex matrices whose product with their transpose is the identity matrix.
3315:
component is represented by block-diagonal matrices with 2-by-2 blocks either
12728:
12706:
12487:, Grundlehren der Mathematischen Wissenschaften, vol. 294, Berlin etc.:
12202:
11761:
11743:
11549:
10618:
representations of the orthogonal groups. (The projective representations of
10075:
6277:
5418:
3001:
2818:
2710:
2504:
1877:
1298:
902:
690:
612:
446:
319:
185:
12651:, Sigma Series in Pure Mathematics, vol. 9, Berlin: Heldermann Verlag,
1598:
under the action of the translations, and all stabilizers are isomorphic to
12403:
10217:
The 'spin' name of the spinor norm can be explained by a connection to the
9265:
When the characteristic is not two, the order of the orthogonal groups are
6749:
6496:
5992:
5812:
4214:
4061:
3990:
2499:
2351:
1896:
1177:
545:
244:
233:
180:
155:
150:
109:
80:
43:
4285:{\displaystyle \operatorname {O} (n)\to \operatorname {O} (n+1)\to S^{n},}
2513:
of absolute value equal to one. This isomorphism sends the complex number
12633:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
12631:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
11728:. In 2 dimensions, the finite groups are either cyclic or dihedral – see
11698:
10651:, the special orthogonal Lie algebra consists of tracefree endomorphisms
10413:
10211:
9754:
In characteristic two, the formulas are the same, except that the factor
6866:
is the number of elements of the finite field (a power of an odd prime).
6703:{\displaystyle \textstyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}
5641:
4338:
4318:
4190:
2666:
2186:
1693:
1193:
968:. The other component consists of all orthogonal matrices of determinant
939:
882:
813:
12124:-frames) are still homogeneous spaces for the orthogonal group, but not
11586:, and consists of the product of the orthogonal group with the group of
7127:{\displaystyle Q={\begin{bmatrix}1&0\\0&-\omega \end{bmatrix}},}
6877:
is not a square in the ground field (that is, if its number of elements
11871:
11790:
10627:
10454:
10389:
is isomorphic to the multiplicative group of the field modulo squares.
10218:
9949:
5988:
5590:
5249:
3994:
3887:
2355:
1391:
1247:
1189:
712:
440:
12140:-frame by an orthogonal map, but this map is not uniquely determined.
9255:{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}.}
8047:{\displaystyle {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},}
6623:{\displaystyle V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,}
3644:
is the subgroup with an even number of minus signs. The Weyl group of
11806:
10871:
Over real numbers, this characterization is used in interpreting the
10457:. One Lie algebra corresponds to both groups. It is often denoted by
10222:
9929:
8478:
The comparison of this proof with the real case may be illuminating.
6644:(that is there is a basis such that the matrix of the restriction of
6336:. Thus, up to isomorphism, there is only one non-degenerate complex
5649:
5308:
4020:
3005:
2713:
1582:
is the vector space of the translations. So, the translations form a
1243:
898:
890:
533:
9177:{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}
8213:{\displaystyle a+\alpha b=(a_{1}+\alpha b_{1})(a_{2}+\alpha b_{2}).}
6209:. So, in the remainder of this section, it is supposed that neither
5179:
11702:
11541:
6553:
can be decomposed as a direct sum of pairwise orthogonal subspaces
5439:
4195:
3943:
2454:
2247:
852:
that preserve a fixed point, where the group operation is given by
839:
70:
12317:
is centerless (but not simply connected), while in even dimension
9786:
is a homomorphism from the orthogonal group to the quotient group
7738:
are two matrices of determinant one in the orthogonal group then
6839:{\displaystyle \textstyle {\begin{bmatrix}\varphi \end{bmatrix}},}
3513:{\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}}
2387:
elementary reflections always suffices to generate any element of
2363:
every rotation can be decomposed into a product of two reflections
12067:
11772:); also equals the intersection of the orthogonal group with the
5723:, these yield vector bundles over the corresponding spheres, and
5466:
in terms of simpler-to-analyze homotopies of lower order. Using π
3244:
412:
326:
12022:
11867:
For the special orthogonal group, the corresponding groups are:
7186:{\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}
2324:{\displaystyle {\begin{bmatrix}-1&0\\0&I\end{bmatrix}},}
2157:{\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}},}
10207:
3906:
2483:
2472:
51:
12707:
John Baez "This Week's Finds in
Mathematical Physics" week 105
6916:
is congruent to 1, modulo 4) the matrix of the restriction of
6881:
is congruent to 3 modulo 4), the matrix of the restriction of
6074:
4193:, this can also be interpreted as a union. On the other hand,
3726:{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}
2907:{\displaystyle {\frac {n(n-1)}{2}}=n^{2}-{\frac {n(n+1)}{2}},}
2371:
is the product of two reflections whose axes form an angle of
856:
transformations. The orthogonal group is sometimes called the
12719:
12184:
11596:
is odd, these two subgroups do not intersect, and they are a
11545:
10142:
6068:
1235:
964:
958:
952:
10626:
are just linear representations of the universal cover, the
6767:
is thus equal to one, and its matrix is congruent either to
2102:
are 2-by-2 rotation matrices, that is matrices of the form
1946:
there is an orthogonal basis, where its matrix has the form
7069:, one can suppose that the matrix of the quadratic form is
6097:, such a form can be written as the difference of a sum of
1552:
where, as usual, the subtraction of two points denotes the
10179:
10010:
of odd characteristic or characteristic zero, which takes
9836:
2 it is equivalent to the determinant: the determinant is
8613:
is the multiplicative group of the element of norm one in
7286:, and thus the orthogonal group consists of the matrices
6798:{\displaystyle \textstyle {\begin{bmatrix}1\end{bmatrix}}}
3872:
The low-dimensional (real) orthogonal groups are familiar
1433:. This stabilizer is (or, more exactly, is isomorphic to)
11160:{\displaystyle \mathrm {O} (2n)\supset \mathrm {USp} (n)}
8475:
and the group of orthogonal matrices of determinant one.
4432:, and one need only to list the lower 8 homotopy groups:
3011:
is a maximal subgroup among those that are isomorphic to
12569:, London Mathematical Society Monographs, vol. 13,
11004:{\displaystyle \mathrm {O} (n)\supset \mathrm {O} (n-1)}
10804:{\displaystyle v\wedge w\mapsto (v,\cdot )w-(w,\cdot )v}
10229:). The spin covering of the orthogonal group provides a
10112:
is a homomorphism from an orthogonal group over a field
9935:
1774:. This implies that the orthogonal group is an internal
997:
such that its inverse equals its transpose is called an
11207:{\displaystyle \mathrm {O} (7)\supset \mathrm {G} _{2}}
11092:
are those that preserve a compatible complex structure
9803:
Algebraically, the
Dickson invariant can be defined as
6526:
Over a field of characteristic different from two, two
6344:, and one associated orthogonal group, usually denoted
12720:
n-dimensional
Special Orthogonal Group parametrization
10546:. Over real numbers, these Lie algebras for different
9917:. Thus in characteristic 2, the determinant is always
9036:
8872:
In the real case, the corresponding isomorphisms are:
8706:
8007:
7776:
7666:
7569:
7301:
7152:
7087:
6937:
6931:
6818:
6812:
6780:
6774:
6668:
6662:
4772:
are identified with stable vector bundles on spheres (
3692:
3373:
3330:
3066:
2559:
2284:
2117:
2024:
1965:
1961:
11966:
11239:
11230:
is also an important subgroup of various Lie groups:
11174:
11116:
11019:
10963:
10820:
10749:
10677:
10657:
10502:
10463:
10246:
10149:
elements), that takes reflection in a vector of norm
9583:
9408:
9274:
9190:
9120:
8977:
8881:
8817:
8766:
8633:
8490:
8383:
8320:
8127:
8001:
7747:
7647:
7550:
7457:
7295:
7140:
7075:
6930:
6811:
6773:
6661:
6562:
6093:, which asserts that, on a vector space of dimension
5872:
4839:
4441:
4226:
4073:
3686:
3555:
3445:
3324:
3197:
3060:
2830:
2553:
2278:
2242:, which asserts that every (non-identity) element of
2111:
1955:
1692:
It follows from this equation that the square of the
1654:
1478:
1317:
1049:
483:
458:
421:
9847:
of the
Dickson invariant and usually has index 2 in
6521:
6460:, these groups are noncompact. As in the real case,
5518:
8310:, and are therefore the image of each other by the
5999:. The homotopy groups that are killed are in turn
3827:
may be too technical for most readers to understand
2821:. Moreover, it can be proved that its dimension is
11996:
11518:
11206:
11159:
11076:
11003:
10848:
10803:
10725:
10663:
10530:
10488:
10307:
9743:
9568:
9393:
9254:
9176:
9103:
8968:is the circle of the complex numbers of norm one;
8953:
8862:
8803:
8749:
8583:
8416:
8358:
8212:
8046:
7984:
7730:
7633:
7533:
7346:
7185:
7126:
6974:
6838:
6797:
6702:
6622:
5968:
5456:Using concrete descriptions of the loop spaces in
5168:
4746:
4284:
4178:
3725:
3623:
3512:
3409:
3232:
3127:
2906:
2754:, that is, the connected component containing the
2646:
2323:
2156:
2073:
1681:
1541:
1353:
1158:
491:
466:
429:
11908:Spin is a 2-to-1 cover, while in even dimension,
10918:are part of a sequence of 8 inclusions used in a
6220:The subgroup of the matrices of determinant 1 in
5180:Computation and interpretation of homotopy groups
4295:which can be understood as "The orthogonal group
2774:can be identified with the group of the matrices
1786:and any subgroup formed with the identity and a
1566:Affine space § Subtraction and Weyl's axioms
12726:
11997:{\displaystyle {\mathfrak {so}}(n,\mathbf {R} )}
6904:is the 2×2 identity matrix. If the dimension of
12357:, but in characteristic 2 these notions differ.
11780:
10920:geometric proof of the Bott periodicity theorem
10392:There is also the connecting homomorphism from
8863:{\displaystyle b={\frac {x-x^{-1}}{2\alpha }}.}
2995:
11701:and can be realized as the symmetry groups of
9924:The Dickson invariant can also be defined for
9832:, Theorem 11.43). Over fields that are not of
5219:-preserving/reversing (this class survives to
3660:by the preimages under the standard injection
12023:Principal homogeneous space: Stiefel manifold
11705:. A very important class of examples are the
10726:{\displaystyle (\varphi A,B)+(A,\varphi B)=0}
10453:matrices, with the Lie bracket given by the
7375:. Moreover, the determinant of the matrix is
6912:is a square in the ground field (that is, if
6295:
5460:, one can interpret the higher homotopies of
3672:of the representatives for the Weyl group of
1016:orthogonal matrices form a subgroup, denoted
793:
12260:Infinite subsets of a compact space have an
10922:, and the corresponding quotient spaces are
10856:for symmetric bilinear forms with signature
10074:in characteristic 2, orthogonal groups over
5374:. There might be a discussion about this on
4056:of the real orthogonal group are related to
4019:and the universal cover corresponds to the
3780:. There might be a discussion about this on
3488:
3478:
3208:
3198:
2933:have the same dimension, and that it has no
2436:
1922:. All these groups are normal subgroups of
1345:
1339:
1333:
1318:
938:. It consists of all orthogonal matrices of
12325:is neither centerless nor simply connected.
10200:over a field can be identified as a Galois
8268:, the image is the multiplicative group of
6996:zero or two. They are denoted respectively
6075:Of indefinite quadratic form over the reals
4189:Since the inclusions are all closed, hence
3867:
3259:acts on the corresponding circle factor of
2791:. Since both members of this equation are
2208:if and only if there are an even number of
1716:. The orthogonal matrices with determinant
1632:
11716:Dimension 3 is particularly studied – see
9885:is commonly defined to be the elements of
8481:Here two group isomorphisms are involved:
5815:, thinking of it as the fundamental group
5677:, and its class in K-theory. Noting that
3233:{\displaystyle \{\pm 1\}^{n}\rtimes S_{n}}
800:
786:
12478:
12476:
12429:
12427:
12412:This Week's Finds in Mathematical Physics
11560:, as exemplified by SSS (side-side-side)
10950:In physics, particularly in the areas of
9873:. Thus when the characteristic is not 2,
5394:Learn how and when to remove this message
5184:
3855:Learn how and when to remove this message
3839:, without removing the technical details.
3800:Learn how and when to remove this message
485:
460:
423:
12485:Quadratic and Hermitian forms over rings
12230:
11916:is a 2-to-1 cover, and in odd dimension
11568:. The group of conformal linear maps of
11096:a compatible symplectic structure – see
10671:which are skew-symmetric for this form (
10378:is essentially the spinor norm, because
8436:. This group is a cyclic group of order
5515:components, and the rest are connected.
4830:fit into the periodicity), one obtains:
4757:
4348:, so the homotopy groups stabilize, and
1269:
877:, where the group operation is given by
12561:
12544:
12237:Representations of classical Lie groups
11924:is a 1-to-1 cover; i.e., isomorphic to
11709:, which include the symmetry groups of
10926:of independent interest – for example,
10180:Galois cohomology and orthogonal groups
8804:{\displaystyle a={\frac {x+x^{-1}}{2}}}
8417:{\displaystyle (a,b)\mapsto a+\alpha b}
6067:) and so on to obtain the higher order
3680:. Those matrices with an odd number of
3251:, where the nontrivial element of each
3028:is the standard one-dimensional torus.
2212:on the diagonal. A pair of eigenvalues
1226:, the elements of the orthogonal group
1168:More generally, given a non-degenerate
14:
12727:
12646:
12597:Classical groups and geometric algebra
12519:
12473:
12467:
12433:
12424:
12148:
11636:is even, these subgroups intersect in
11544:, real orthogonal transforms preserve
9932:in a similar way (in all dimensions).
9829:
9796:(integers modulo 2), taking the value
6988:This implies that if the dimension of
6288:corresponds to space coordinates, and
4766:, homotopy groups of the stable space
2761:
2490:. The orientation-preserving subgroup
2365:. More precisely, a rotation of angle
1664:
1136:
1115:
358:Classification of finite simple groups
12594:
12532:
11692:
11108:also preserves a complex orientation.
10849:{\displaystyle {\mathfrak {so}}(p,q)}
10641:More generally, given a vector space
10531:{\displaystyle {\mathfrak {so}}(n,F)}
10096:, acted upon by the orthogonal group.
9936:Orthogonal groups of characteristic 2
7061:For studying the orthogonal group of
6516:
3837:make it understandable to non-experts
2224:can be identified with a rotation by
2216:can be identified with a rotation by
12649:The Geometry of the Classical Groups
12628:
12538:
12482:
12390:
12378:
12366:
12353:is equivalent to that in terms of a
12349:not 2, the definition in terms of a
12155:Coordinate rotations and reflections
12008:is the simply connected form, while
10489:{\displaystyle {\mathfrak {o}}(n,F)}
10329:algebraic group of square roots of 1
9944:, but this term is no longer used.)
9843:The special orthogonal group is the
9777:
6430:are complex Lie groups of dimension
6292:corresponds to the time coordinate.
5346:
3957:
3811:
3752:
2427:that is not a product of fewer than
1744:, which are those that preserve the
1645:, which are the matrices such that
1594:, the stabilizers of two points are
1542:{\displaystyle p(g)(y-x)=g(y)-g(x),}
12191:
11972:
11969:
11956:are Lie group forms of the compact
11890:Projective special orthogonal group
10826:
10823:
10508:
10505:
10466:
9865:is not 2, the Dickson Invariant is
8279:, which is a cyclic group of order
7193:belongs to the orthogonal group if
6499:, whereas the fundamental group of
4401:, therefore the homotopy groups of
1682:{\displaystyle QQ^{\mathsf {T}}=I.}
1609:Moreover, the Euclidean group is a
864:. Equivalently, it is the group of
840:distance-preserving transformations
24:
12603:, vol. 39, Providence, R.I.:
12213:
12165:
11529:
11499:
11481:
11462:
11444:
11425:
11407:
11388:
11370:
11351:
11333:
11314:
11290:
11287:
11284:
11266:
11245:
11217:
11194:
11176:
11144:
11141:
11138:
11118:
11061:
11058:
11041:
11021:
10982:
10965:
10740:. The correspondence is given by:
10638:, which are important in physics.
10293:
10289:
10274:
10271:
10268:
10103:
9590:
9415:
9280:
5951:
5854:
4245:
4227:
4161:
4156:
4110:
4092:
4074:
4034:
2688:
2509:, the multiplicative group of the
1050:
908:The orthogonal group in dimension
25:
12761:
12682:
12134:-frame can be taken to any other
11726:list of spherical symmetry groups
10878:
10634:).) The latter are the so-called
6522:Characteristic different from two
6193:The standard orthogonal group is
5519:Interpretation of homotopy groups
5342:
4321:) is the orthogonal group of the
2544:to the special orthogonal matrix
2419:) is an example of an element of
1933:
942:1. This group is also called the
12402:
11987:
11785:The orthogonal group is neither
11735:Other finite subgroups include:
11718:point groups in three dimensions
10945:
10398:of the orthogonal group, to the
9010:
8983:
8903:
8887:
8674:
8639:
8521:
8496:
8443:which consists of the powers of
5611:topological quantum field theory
5351:
5050:
4976:
4963:
4925:
4912:
4874:
4776:), with a dimension shift of 1:
4736:
4602:
4534:
4521:
4486:
4473:
3816:
3757:
2250:about a unique axis–angle pair.
1895:). Its finite subgroups are the
1408:is the subgroup of the elements
50:
12601:Graduate Studies in Mathematics
12525:
12299:
12267:
12254:
12118:orthonormal bases (orthonormal
10556:of two of the four families of
8359:{\displaystyle y=x^{-1}=x^{q},}
7498:
7473:
5859:The orthogonal group anchors a
4064:of the sequence of inclusions:
3733:blocks have no remaining final
3424:chosen to make the determinant
3367:
3361:
2963:nonsingular algebraic varieties
1354:{\displaystyle \|g(x)\|=\|x\|.}
12512:
12460:
12396:
12384:
12372:
12360:
12339:
11991:
11977:
11958:special orthogonal Lie algebra
11730:point groups in two dimensions
11509:
11503:
11472:
11466:
11435:
11429:
11398:
11392:
11361:
11355:
11324:
11318:
11303:
11294:
11276:
11270:
11255:
11249:
11186:
11180:
11154:
11148:
11131:
11122:
11071:
11065:
11051:
11045:
11034:
11025:
10998:
10986:
10975:
10969:
10843:
10831:
10795:
10783:
10774:
10762:
10759:
10714:
10699:
10693:
10678:
10544:special orthogonal Lie algebra
10525:
10513:
10483:
10471:
10407:
10299:
10284:
10263:
10250:
9651:
9639:
9617:
9602:
9476:
9464:
9442:
9427:
9307:
9286:
9028:
9014:
9000:
8991:
8928:
8911:
8698:
8684:
8663:
8647:
8566:
8554:
8546:
8529:
8517:
8505:
8399:
8396:
8384:
8204:
8175:
8172:
8143:
7390:. This square root belongs to
5963:
5957:
5948:
5945:
5939:
5930:
5927:
5921:
5912:
5909:
5903:
5894:
5891:
5885:
5876:
5451:
5149:
5140:
5113:
5104:
5077:
5068:
5039:
5030:
5003:
4994:
4952:
4943:
4901:
4892:
4863:
4854:
4725:
4719:
4692:
4686:
4659:
4653:
4626:
4620:
4591:
4585:
4558:
4552:
4510:
4504:
4462:
4456:
4266:
4263:
4251:
4242:
4239:
4233:
4173:
4167:
4122:
4116:
4104:
4098:
4086:
4080:
3595:
2892:
2880:
2849:
2837:
2630:
2624:
2613:
2607:
2594:
2588:
2574:
2568:
2253:
1704:, and thus the determinant of
1533:
1527:
1518:
1512:
1503:
1491:
1488:
1482:
1330:
1324:
1103:
1091:
1068:
1056:
893:). The orthogonal group is an
719:Infinite dimensional Lie group
13:
1:
12605:American Mathematical Society
12554:
12160:Reflection through the origin
11864:These are all 2-to-1 covers.
10346:, which is simply the group
9859:. When the characteristic of
7995:This is an orthogonal matrix
6472:is not simply connected: For
6087:nondegenerate quadratic forms
5985:Eilenberg–MacLane space
5302:
4407:are 8-fold periodic, meaning
2407:reflection through the origin
1918:, for every positive integer
1805:is the identity matrix) is a
12332:
12220:list of finite simple groups
12090:The other Stiefel manifolds
12012:is the centerless form, and
11781:Covering and quotient groups
11564:and AAA (angle-angle-angle)
10416:corresponding to Lie groups
9869:whenever the determinant is
6547:Witt's decomposition theorem
5298:; this latter thus vanishes.
4317:of a point (thought of as a
4213:, and one has the following
3545:of the product homomorphism
2996:Maximal tori and Weyl groups
2929:. This implies that all its
1793:The group with two elements
1186:orthogonal group of the form
975:By extension, for any field
918:. The one that contains the
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
27:Type of group in mathematics
7:
12695:Encyclopedia of Mathematics
12198:indefinite orthogonal group
12143:
12035:principal homogeneous space
11846:projective orthogonal group
11758:Signed permutation matrices
9782:For orthogonal groups, the
7401:if the orthogonal group is
6358:complex orthogonal matrices
6318:variables is equivalent to
6081:Indefinite orthogonal group
3748:
1720:form a subgroup called the
1188:is the group of invertible
881:(an orthogonal matrix is a
217:List of group theory topics
10:
12766:
12647:Taylor, Donald E. (1992),
12434:Wilson, Robert A. (2009).
12026:
11584:conformal orthogonal group
11533:
10580:, while in even dimension
8598:is a primitive element of
6985:is any non-square scalar.
6710:), and the restriction of
6296:Of complex quadratic forms
6091:Sylvester's law of inertia
6078:
5995:, and are preceded by the
5811:can be interpreted as the
5784:From the point of view of
4058:homotopy groups of spheres
3270:, and the symmetric group
2220:and a pair of eigenvalues
1199:All orthogonal groups are
12483:Knus, Max-Albert (1991),
12225:list of simple Lie groups
12037:for the orthogonal group
11658:Similarly one can define
10000:is the bilinear form and
6763:is odd, the dimension of
6746:Chevalley–Warning theorem
5307:From general facts about
4015:the fundamental group is
2437:Symmetry group of spheres
2269:whose canonical form is
1305:; that is, endomorphisms
12595:Grove, Larry C. (2002),
12567:Rational Quadratic Forms
12535:, Theorem 6.6 and 14.16)
12436:The finite simple groups
12247:
12128:homogeneous spaces: any
10664:{\displaystyle \varphi }
8223:It follows that the map
5665:tautological line bundle
4323:perpendicular complement
4027:is the unique connected
3868:Low-dimensional topology
3420:with the last component
2400:Cartan–Dieudonné theorem
2240:Euler's rotation theorem
1758:is a normal subgroup of
1722:special orthogonal group
1633:Special orthogonal group
928:special orthogonal group
858:general orthogonal group
335:Elementary abelian group
212:Glossary of group theory
12750:Linear algebraic groups
12629:Hall, Brian C. (2015),
12351:symmetric bilinear form
12016:is in general neither.
11566:similarity of triangles
11562:congruence of triangles
10940:Lagrangian Grassmannian
10558:semisimple Lie algebras
10333:connecting homomorphism
6889:is congruent to either
6310:, every non-degenerate
6280:that is fundamental in
6085:Over the real numbers,
5825:Lagrangian Grassmannian
3049:block-diagonal matrices
2200:The element belongs to
1811:characteristic subgroup
1282:is the subgroup of the
1266:to orthogonal vectors.
1206:
1170:symmetric bilinear form
991:matrix with entries in
12712:John Baez on Octonions
11998:
11793:, and thus has both a
11520:
11208:
11161:
11078:
11005:
10850:
10805:
10727:
10665:
10540:orthogonal Lie algebra
10532:
10490:
10309:
10008:Householder reflection
9745:
9708:
9570:
9533:
9395:
9358:
9256:
9178:
9105:
8955:
8864:
8805:
8751:
8585:
8418:
8360:
8312:Frobenius automorphism
8214:
8048:
7986:
7732:
7635:
7535:
7348:
7187:
7128:
6976:
6840:
6799:
6704:
6624:
6534:if their matrices are
5970:
5290:, which surjects onto
5185:Low-dimensional groups
5170:
4764:clutching construction
4748:
4286:
4180:
4160:
3727:
3625:
3514:
3411:
3234:
3129:
2965:of the same dimension
2931:irreducible components
2908:
2648:
2325:
2181:This results from the
2158:
2075:
1683:
1543:
1469:, which is defined by
1355:
1213:Euclidean vector space
1160:
999:orthogonal matrix over
860:, by analogy with the
751:Linear algebraic group
493:
468:
431:
12264:and are not discrete.
12231:Representation theory
11999:
11707:finite Coxeter groups
11521:
11222:The orthogonal group
11209:
11162:
11079:
11006:
10851:
10806:
10728:
10666:
10533:
10491:
10310:
9746:
9682:
9571:
9507:
9396:
9338:
9257:
9179:
9106:
8956:
8865:
8806:
8752:
8586:
8419:
8361:
8215:
8049:
7987:
7733:
7636:
7536:
7349:
7188:
7129:
7023:The orthogonal group
6977:
6841:
6800:
6748:asserts that, over a
6705:
6625:
6363:As in the real case,
6356:. It is the group of
6101:squares and a sum of
5981:short exact sequences
5971:
5483:have two components,
5323:always vanishes, and
5171:
4758:Relation to KO-theory
4749:
4287:
4181:
4140:
3728:
3626:
3515:
3412:
3311:on the diagonal. The
3235:
3130:
2976:. The component with
2927:complete intersection
2909:
2766:The orthogonal group
2657:In higher dimension,
2649:
2441:The orthogonal group
2326:
2159:
2076:
1684:
1544:
1356:
1270:In Euclidean geometry
1161:
879:matrix multiplication
494:
469:
432:
12745:Euclidean symmetries
11964:
11740:Permutation matrices
11628:multiplicative group
11237:
11172:
11114:
11017:
10961:
10818:
10747:
10675:
10655:
10500:
10461:
10364:-valued points, to
10244:
10231:short exact sequence
10206:, or twisted forms (
10134:multiplicative group
9581:
9406:
9272:
9188:
9118:
8975:
8879:
8815:
8764:
8631:
8488:
8381:
8318:
8125:
7999:
7745:
7645:
7548:
7455:
7293:
7138:
7073:
6928:
6869:If the dimension of
6809:
6771:
6759:If the dimension of
6659:
6560:
6453:is twice that). For
6447:(the dimension over
5870:
5364:confusing or unclear
4837:
4439:
4224:
4071:
3770:confusing or unclear
3684:
3553:
3443:
3322:
3195:
3058:
2828:
2740:connected components
2551:
2494:is isomorphic (as a
2276:
2261:are the elements of
2231:The special case of
2109:
1953:
1888:is not abelian when
1732:, consisting of all
1652:
1476:
1373:Euclidean isometries
1371:be the group of the
1315:
1297:, consisting of all
1284:general linear group
1047:
1030:general linear group
916:connected components
862:general linear group
481:
456:
419:
12345:For base fields of
12149:Specific transforms
12083:basis to any other
11098:2-out-of-3 property
10636:spin representation
10560:: in odd dimension
10221:(more accurately a
7420:otherwise. Setting
6752:, the dimension of
5786:symplectic geometry
5623:be any of the four
5372:clarify the section
4307:on the unit sphere
3778:clarify the section
2917:which implies that
2762:As algebraic groups
2471:, this is just the
2381:A product of up to
2084:where the matrices
1938:For any element of
1643:orthogonal matrices
1444:There is a natural
1396:stabilizer subgroup
875:orthogonal matrices
125:Group homomorphisms
35:Algebraic structure
12690:"Orthogonal group"
12305:In odd dimension,
12262:accumulation point
11994:
11693:Discrete subgroups
11666:; this is always:
11516:
11514:
11204:
11157:
11074:
11011:– preserve an axis
11001:
10846:
10801:
10723:
10661:
10554:compact real forms
10528:
10486:
10305:
10145:multiplication by
10066:In odd dimensions
9942:hypoabelian groups
9901:. Each element in
9741:
9566:
9391:
9252:
9174:
9101:
9099:
9088:
8951:
8949:
8860:
8801:
8747:
8745:
8734:
8581:
8579:
8414:
8356:
8314:. This meant that
8210:
8044:
8035:
7982:
7973:
7728:
7722:
7631:
7625:
7531:
7344:
7335:
7183:
7177:
7124:
7115:
7059:
6972:
6971:
6962:
6836:
6835:
6826:
6795:
6794:
6788:
6733:for every nonzero
6700:
6699:
6693:
6620:
6517:Over finite fields
6089:are classified by
5966:
5166:
5164:
4744:
4742:
4282:
4176:
3964:algebraic topology
3723:
3717:
3652:is represented in
3621:
3510:
3431:The Weyl group of
3407:
3398:
3355:
3242:elementary abelian
3230:
3190:semidirect product
3125:
3116:
2935:embedded component
2904:
2793:symmetric matrices
2752:identity component
2683:topological spaces
2669:structures of the
2644:
2635:
2498:Lie group) to the
2361:In dimension two,
2354:with respect to a
2321:
2312:
2154:
2145:
2071:
2062:
2058:
2007:
1776:semidirect product
1679:
1611:semidirect product
1563:(for details, see
1539:
1446:group homomorphism
1351:
1301:that preserve the
1264:orthogonal vectors
1156:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
12614:978-0-8218-2019-3
12445:978-1-84800-987-5
12393:Proposition 13.10
12064:orthonormal bases
11722:polyhedral groups
11711:regular polytopes
10538:, and called the
10227:Clifford algebras
10186:Galois cohomology
10184:In the theory of
10080:symplectic groups
9897:with determinant
9828:is the identity (
9784:Dickson invariant
9778:Dickson invariant
9774:must be removed.
9247:
9172:
8855:
8799:
8455:primitive element
8373:. For every such
7526:
7496:
7057:
6497:cyclic of order 2
6481:fundamental group
6282:relativity theory
6135:entries equal to
6127:entries equal to
5983:starting with an
5625:division algebras
5404:
5403:
5396:
5223:and hence stably)
4774:up to isomorphism
4203:homogeneous space
3991:cyclic of order 2
3975:fundamental group
3958:Fundamental group
3865:
3864:
3857:
3810:
3809:
3802:
3365:
3173:on the diagonal.
2899:
2856:
2809:This proves that
2191:complex conjugate
1827:is even, also of
1734:direct isometries
1639:orthonormal basis
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
12757:
12718:
12703:
12677:
12643:
12625:
12591:
12548:
12542:
12536:
12529:
12523:
12516:
12510:
12509:
12480:
12471:
12464:
12458:
12457:
12431:
12422:
12421:
12419:
12418:
12400:
12394:
12388:
12382:
12376:
12370:
12364:
12358:
12343:
12326:
12324:
12316:
12303:
12297:
12295:
12291:
12271:
12265:
12258:
12208:symplectic group
12187:
12179:
12171:rotation group,
12139:
12133:
12123:
12113:
12103:
12072:
12061:
12047:Stiefel manifold
12044:
12029:Stiefel manifold
12015:
12011:
12007:
12003:
12001:
12000:
11995:
11990:
11976:
11975:
11955:
11947:
11939:
11932:. These groups,
11931:
11923:
11915:
11903:
11885:
11859:
11840:
11824:
11801:, respectively:
11787:simply connected
11774:integer matrices
11771:
11753:
11688:
11665:
11654:
11639:
11635:
11625:
11614:
11595:
11581:
11573:
11525:
11523:
11522:
11517:
11515:
11502:
11490:
11489:
11484:
11465:
11453:
11452:
11447:
11428:
11416:
11415:
11410:
11391:
11379:
11378:
11373:
11354:
11342:
11341:
11336:
11317:
11293:
11269:
11248:
11229:
11213:
11211:
11210:
11205:
11203:
11202:
11197:
11179:
11166:
11164:
11163:
11158:
11147:
11121:
11107:
11091:
11083:
11081:
11080:
11075:
11064:
11044:
11024:
11010:
11008:
11007:
11002:
10985:
10968:
10937:
10924:symmetric spaces
10917:
10901:
10867:
10855:
10853:
10852:
10847:
10830:
10829:
10810:
10808:
10807:
10802:
10739:
10732:
10730:
10729:
10724:
10670:
10668:
10667:
10662:
10650:
10646:
10625:
10609:
10602:Since the group
10598:
10588:
10579:
10568:
10551:
10537:
10535:
10534:
10529:
10512:
10511:
10495:
10493:
10492:
10487:
10470:
10469:
10452:
10440:consists of the
10439:
10427:
10403:
10397:
10388:
10377:
10363:
10357:
10345:
10326:
10314:
10312:
10311:
10306:
10298:
10297:
10296:
10283:
10282:
10277:
10262:
10261:
10235:algebraic groups
10205:
10190:algebraic groups
10171:
10160:
10155:to the image of
10154:
10141:
10131:
10117:
10095:
10088:
10078:are the same as
10073:
10062:
10045:
10015:
10005:
9999:
9993:
9963:
9957:
9920:
9916:
9913:has determinant
9912:
9900:
9896:
9884:
9872:
9868:
9864:
9858:
9839:
9827:
9821:
9799:
9795:
9773:
9771:
9757:
9750:
9748:
9747:
9742:
9737:
9733:
9726:
9725:
9707:
9696:
9681:
9677:
9670:
9669:
9655:
9654:
9624:
9620:
9598:
9597:
9575:
9573:
9572:
9567:
9562:
9558:
9551:
9550:
9532:
9521:
9506:
9502:
9495:
9494:
9480:
9479:
9449:
9445:
9423:
9422:
9400:
9398:
9397:
9392:
9387:
9383:
9376:
9375:
9357:
9352:
9337:
9336:
9335:
9334:
9314:
9310:
9261:
9259:
9258:
9253:
9248:
9246:
9238:
9237:
9236:
9218:
9217:
9204:
9183:
9181:
9180:
9175:
9173:
9168:
9167:
9166:
9148:
9147:
9134:
9110:
9108:
9107:
9102:
9100:
9093:
9092:
9013:
8986:
8967:
8960:
8958:
8957:
8952:
8950:
8943:
8942:
8906:
8895:
8890:
8869:
8867:
8866:
8861:
8856:
8854:
8846:
8845:
8844:
8825:
8810:
8808:
8807:
8802:
8800:
8795:
8794:
8793:
8774:
8756:
8754:
8753:
8748:
8746:
8739:
8738:
8683:
8682:
8677:
8659:
8658:
8642:
8623:
8612:
8608:
8597:
8590:
8588:
8587:
8582:
8580:
8573:
8572:
8524:
8504:
8499:
8474:
8467:
8452:
8448:
8442:
8431:
8423:
8421:
8420:
8415:
8376:
8372:
8365:
8363:
8362:
8357:
8352:
8351:
8339:
8338:
8305:
8301:
8297:
8282:
8278:
8267:
8252:
8241:
8219:
8217:
8216:
8211:
8203:
8202:
8187:
8186:
8171:
8170:
8155:
8154:
8117:
8085:
8053:
8051:
8050:
8045:
8040:
8039:
7991:
7989:
7988:
7983:
7978:
7977:
7970:
7969:
7960:
7959:
7947:
7946:
7937:
7936:
7922:
7921:
7912:
7911:
7896:
7895:
7886:
7885:
7869:
7868:
7859:
7858:
7846:
7845:
7836:
7835:
7824:
7823:
7814:
7813:
7798:
7797:
7788:
7787:
7767:
7766:
7757:
7756:
7737:
7735:
7734:
7729:
7727:
7726:
7719:
7718:
7707:
7706:
7690:
7689:
7678:
7677:
7657:
7656:
7640:
7638:
7637:
7632:
7630:
7629:
7622:
7621:
7610:
7609:
7593:
7592:
7581:
7580:
7560:
7559:
7540:
7538:
7537:
7532:
7527:
7525:
7517:
7506:
7497:
7492:
7481:
7447:
7433:
7419:
7408:
7400:
7389:
7385:
7378:
7374:
7367:
7353:
7351:
7350:
7345:
7340:
7339:
7285:
7278:
7268:
7258:
7247:
7243:
7239:
7235:
7221:
7210:
7199:
7192:
7190:
7189:
7184:
7182:
7181:
7133:
7131:
7130:
7125:
7120:
7119:
7068:
7053:
7046:
7030:
7019:
7007:
6995:
6991:
6984:
6981:
6979:
6978:
6973:
6967:
6966:
6924:is congruent to
6923:
6919:
6915:
6911:
6907:
6903:
6899:
6892:
6888:
6884:
6880:
6876:
6872:
6865:
6861:
6849:
6845:
6843:
6842:
6837:
6831:
6830:
6804:
6802:
6801:
6796:
6793:
6792:
6766:
6762:
6756:is at most two.
6755:
6740:
6736:
6732:
6717:
6713:
6709:
6707:
6706:
6701:
6698:
6697:
6654:
6647:
6642:hyperbolic plane
6639:
6629:
6627:
6626:
6621:
6610:
6609:
6591:
6590:
6578:
6577:
6552:
6545:More precisely,
6512:
6506:
6494:
6478:
6471:
6459:
6452:
6446:
6440:
6429:
6417:
6402:
6391:; it is denoted
6390:
6378:
6374:
6355:
6343:
6335:
6317:
6305:
6291:
6287:
6275:
6268:
6255:
6243:
6231:
6216:
6212:
6208:
6189:
6169:
6150:
6138:
6134:
6130:
6126:
6118:
6104:
6100:
6096:
6059:
6040:
6021:
6002:
5975:
5973:
5972:
5967:
5850:
5834:
5822:
5810:
5780:is generated by
5779:
5766:is generated by
5765:
5752:is generated by
5751:
5738:is generated by
5737:
5722:
5712:
5702:
5689:
5676:
5662:
5653:
5645:
5637:
5631:
5622:
5608:
5588:
5564:
5540:
5510:
5496:
5482:
5475:
5465:
5458:Bott periodicity
5443:
5426:
5416:
5399:
5392:
5388:
5385:
5379:
5355:
5354:
5347:
5334:
5322:
5297:
5289:
5271:
5247:
5222:
5214:
5175:
5173:
5172:
5167:
5165:
5139:
5138:
5103:
5102:
5067:
5066:
5053:
5029:
5028:
4993:
4992:
4979:
4971:
4966:
4942:
4941:
4928:
4920:
4915:
4891:
4890:
4877:
4853:
4852:
4829:
4822:
4800:
4771:
4753:
4751:
4750:
4745:
4743:
4739:
4718:
4717:
4685:
4684:
4652:
4651:
4619:
4618:
4605:
4584:
4583:
4551:
4550:
4537:
4529:
4524:
4503:
4502:
4489:
4481:
4476:
4455:
4454:
4431:
4406:
4400:
4388:Bott periodicity
4382:
4371:
4345:
4336:
4312:
4302:
4291:
4289:
4288:
4283:
4278:
4277:
4212:
4200:
4185:
4183:
4182:
4177:
4159:
4154:
4055:
4026:
4014:
4003:
3988:
3972:
3952:
3932:
3927:
3918:
3911:
3903:
3898:
3885:
3860:
3853:
3849:
3846:
3840:
3820:
3819:
3812:
3805:
3798:
3794:
3791:
3785:
3761:
3760:
3753:
3744:
3736:
3732:
3730:
3729:
3724:
3722:
3721:
3679:
3671:
3659:
3651:
3643:
3630:
3628:
3627:
3622:
3620:
3619:
3607:
3606:
3594:
3590:
3589:
3588:
3570:
3569:
3548:
3540:
3527:
3519:
3517:
3516:
3511:
3509:
3508:
3496:
3495:
3474:
3473:
3461:
3460:
3439:is the subgroup
3438:
3427:
3423:
3416:
3414:
3413:
3408:
3403:
3402:
3366:
3363:
3360:
3359:
3314:
3310:
3306:
3297:
3289:
3282:
3278:
3265:
3258:
3254:
3239:
3237:
3236:
3231:
3229:
3228:
3216:
3215:
3187:
3172:
3168:
3160:
3152:
3148:
3134:
3132:
3131:
3126:
3121:
3120:
3113:
3112:
3102:
3094:
3088:
3080:
3078:
3077:
3046:
3038:
3027:
3020:
3016:
2991:
2983:
2975:
2960:
2952:
2944:
2924:
2913:
2911:
2910:
2905:
2900:
2895:
2875:
2870:
2869:
2857:
2852:
2832:
2816:
2805:
2795:, this provides
2790:
2777:
2773:
2749:
2737:
2729:
2708:
2700:
2680:
2672:
2664:
2653:
2651:
2650:
2645:
2640:
2639:
2543:
2535:
2508:
2502:, also known as
2493:
2489:
2470:
2461:
2448:
2432:
2426:
2418:
2394:
2386:
2377:
2370:
2349:
2337:
2330:
2328:
2327:
2322:
2317:
2316:
2268:
2245:
2237:
2227:
2223:
2219:
2215:
2211:
2207:
2196:
2183:spectral theorem
2177:
2163:
2161:
2160:
2155:
2150:
2149:
2101:
2080:
2078:
2077:
2072:
2067:
2066:
2059:
2048:
2047:
2044:
2038:
2035:
2034:
2008:
2004:
2003:
1993:
1992:
1989:
1983:
1980:
1979:
1977:
1976:
1945:
1929:
1925:
1921:
1915:
1908:
1894:
1887:
1875:
1868:
1860:
1849:is the internal
1848:
1840:
1834:
1826:
1820:
1804:
1800:
1785:
1773:
1765:
1757:
1743:
1731:
1719:
1715:
1711:
1707:
1703:
1699:
1688:
1686:
1685:
1680:
1669:
1668:
1667:
1628:
1620:
1605:
1593:
1581:
1562:
1548:
1546:
1545:
1540:
1468:
1460:
1452:
1440:
1432:
1418:
1407:
1389:
1383:
1370:
1360:
1358:
1357:
1352:
1310:
1296:
1281:
1261:
1255:
1233:
1225:
1219:
1201:algebraic groups
1165:
1163:
1162:
1157:
1152:
1148:
1141:
1140:
1139:
1120:
1119:
1118:
1042:
1027:
1015:
1005:
996:
990:
980:
971:
967:
961:
955:
937:
920:identity element
913:
873:
851:
833:
825:
818:orthogonal group
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
21:
12765:
12764:
12760:
12759:
12758:
12756:
12755:
12754:
12740:Quadratic forms
12725:
12724:
12716:
12688:
12685:
12680:
12659:
12641:
12615:
12581:
12563:Cassels, J.W.S.
12557:
12552:
12551:
12543:
12539:
12530:
12526:
12517:
12513:
12499:
12491:, p. 224,
12489:Springer-Verlag
12481:
12474:
12465:
12461:
12446:
12432:
12425:
12416:
12414:
12401:
12397:
12389:
12385:
12377:
12373:
12365:
12361:
12344:
12340:
12335:
12330:
12329:
12318:
12306:
12304:
12300:
12293:
12273:
12272:
12268:
12259:
12255:
12250:
12233:
12216:
12214:Lists of groups
12194:
12183:
12173:
12168:
12166:Specific groups
12151:
12146:
12135:
12129:
12119:
12105:
12096:
12091:
12068:
12054:
12049:
12038:
12031:
12025:
12013:
12009:
12005:
11986:
11968:
11967:
11965:
11962:
11961:
11949:
11941:
11933:
11925:
11917:
11909:
11893:
11875:
11849:
11830:
11826:
11814:
11810:
11783:
11770:
11764:
11752:
11746:
11695:
11667:
11659:
11641:
11637:
11631:
11616:
11601:
11591:
11575:
11569:
11548:, and are thus
11538:
11536:Conformal group
11532:
11530:Conformal group
11513:
11512:
11498:
11491:
11485:
11480:
11479:
11476:
11475:
11461:
11454:
11448:
11443:
11442:
11439:
11438:
11424:
11417:
11411:
11406:
11405:
11402:
11401:
11387:
11380:
11374:
11369:
11368:
11365:
11364:
11350:
11343:
11337:
11332:
11331:
11328:
11327:
11313:
11306:
11283:
11280:
11279:
11265:
11258:
11244:
11240:
11238:
11235:
11234:
11223:
11220:
11218:Lie supergroups
11198:
11193:
11192:
11175:
11173:
11170:
11169:
11137:
11117:
11115:
11112:
11111:
11101:
11085:
11057:
11040:
11020:
11018:
11015:
11014:
10981:
10964:
10962:
10959:
10958:
10948:
10927:
10903:
10887:
10886:The inclusions
10881:
10857:
10822:
10821:
10819:
10816:
10815:
10748:
10745:
10744:
10734:
10676:
10673:
10672:
10656:
10653:
10652:
10648:
10642:
10619:
10603:
10590:
10587:
10581:
10570:
10567:
10561:
10547:
10504:
10503:
10501:
10498:
10497:
10465:
10464:
10462:
10459:
10458:
10444:
10429:
10417:
10410:
10399:
10393:
10386:
10379:
10375:
10365:
10359:
10351:
10347:
10343:
10336:
10325:
10319:
10292:
10288:
10287:
10278:
10267:
10266:
10257:
10253:
10245:
10242:
10241:
10201:
10198:quadratic forms
10182:
10162:
10156:
10150:
10137:
10122:
10113:
10106:
10104:The spinor norm
10090:
10083:
10067:
10054:
10017:
10011:
10001:
9995:
9965:
9959:
9958:takes a vector
9953:
9938:
9926:Clifford groups
9918:
9914:
9902:
9898:
9886:
9874:
9870:
9866:
9860:
9848:
9837:
9823:
9804:
9797:
9787:
9780:
9761:
9759:
9755:
9718:
9714:
9713:
9709:
9697:
9686:
9665:
9661:
9660:
9656:
9635:
9631:
9593:
9589:
9588:
9584:
9582:
9579:
9578:
9543:
9539:
9538:
9534:
9522:
9511:
9490:
9486:
9485:
9481:
9460:
9456:
9418:
9414:
9413:
9409:
9407:
9404:
9403:
9368:
9364:
9363:
9359:
9353:
9342:
9330:
9326:
9325:
9321:
9279:
9275:
9273:
9270:
9269:
9263:
9239:
9226:
9222:
9210:
9206:
9205:
9203:
9189:
9186:
9185:
9156:
9152:
9140:
9136:
9135:
9133:
9119:
9116:
9115:
9098:
9097:
9087:
9086:
9075:
9060:
9059:
9048:
9032:
9031:
9024:
9018:
9017:
9009:
8987:
8982:
8978:
8976:
8973:
8972:
8965:
8948:
8947:
8935:
8931:
8924:
8918:
8917:
8907:
8902:
8891:
8886:
8882:
8880:
8877:
8876:
8847:
8837:
8833:
8826:
8824:
8816:
8813:
8812:
8786:
8782:
8775:
8773:
8765:
8762:
8761:
8744:
8743:
8733:
8732:
8727:
8718:
8717:
8712:
8702:
8701:
8694:
8688:
8687:
8678:
8673:
8672:
8654:
8650:
8643:
8638:
8634:
8632:
8629:
8628:
8622:
8614:
8610:
8607:
8599:
8595:
8578:
8577:
8553:
8549:
8542:
8536:
8535:
8525:
8520:
8500:
8495:
8491:
8489:
8486:
8485:
8472:
8466:
8458:
8450:
8444:
8437:
8425:
8382:
8379:
8378:
8374:
8367:
8347:
8343:
8331:
8327:
8319:
8316:
8315:
8303:
8299:
8287:
8286:In the case of
8280:
8277:
8269:
8257:
8256:In the case of
8251:
8243:
8224:
8198:
8194:
8182:
8178:
8166:
8162:
8150:
8146:
8126:
8123:
8122:
8116:
8110:
8103:
8097:
8087:
8084:
8078:
8071:
8065:
8055:
8034:
8033:
8028:
8019:
8018:
8013:
8003:
8002:
8000:
7997:
7996:
7972:
7971:
7965:
7961:
7955:
7951:
7942:
7938:
7932:
7928:
7923:
7917:
7913:
7907:
7903:
7891:
7887:
7881:
7877:
7871:
7870:
7864:
7860:
7854:
7850:
7841:
7837:
7831:
7827:
7825:
7819:
7815:
7809:
7805:
7793:
7789:
7783:
7779:
7772:
7771:
7762:
7758:
7752:
7748:
7746:
7743:
7742:
7721:
7720:
7714:
7710:
7708:
7702:
7698:
7692:
7691:
7685:
7681:
7679:
7673:
7669:
7662:
7661:
7652:
7648:
7646:
7643:
7642:
7624:
7623:
7617:
7613:
7611:
7605:
7601:
7595:
7594:
7588:
7584:
7582:
7576:
7572:
7565:
7564:
7555:
7551:
7549:
7546:
7545:
7518:
7507:
7505:
7482:
7480:
7456:
7453:
7452:
7435:
7421:
7418:
7410:
7402:
7399:
7391:
7387:
7383:
7376:
7369:
7358:
7334:
7333:
7325:
7313:
7312:
7307:
7297:
7296:
7294:
7291:
7290:
7280:
7270:
7260:
7257:
7249:
7245:
7241:
7237:
7223:
7212:
7201:
7194:
7176:
7175:
7170:
7164:
7163:
7158:
7148:
7147:
7139:
7136:
7135:
7114:
7113:
7105:
7099:
7098:
7093:
7083:
7082:
7074:
7071:
7070:
7062:
7048:
7036:
7024:
7009:
6997:
6993:
6989:
6982:
6961:
6960:
6955:
6949:
6948:
6943:
6933:
6932:
6929:
6926:
6925:
6921:
6917:
6913:
6909:
6905:
6901:
6894:
6890:
6886:
6882:
6878:
6874:
6870:
6863:
6851:
6847:
6825:
6824:
6814:
6813:
6810:
6807:
6806:
6787:
6786:
6776:
6775:
6772:
6769:
6768:
6764:
6760:
6753:
6738:
6734:
6723:
6715:
6711:
6692:
6691:
6686:
6680:
6679:
6674:
6664:
6663:
6660:
6657:
6656:
6653:
6649:
6645:
6638:
6634:
6605:
6601:
6586:
6582:
6573:
6569:
6561:
6558:
6557:
6550:
6528:quadratic forms
6524:
6519:
6508:
6500:
6484:
6473:
6461:
6454:
6448:
6442:
6431:
6419:
6407:
6392:
6380:
6376:
6364:
6345:
6341:
6338:quadratic space
6334:
6325:
6319:
6315:
6308:complex numbers
6301:
6300:Over the field
6298:
6289:
6285:
6273:
6258:
6245:
6233:
6221:
6214:
6210:
6194:
6171:
6159:
6140:
6136:
6132:
6128:
6124:
6121:diagonal matrix
6106:
6102:
6098:
6094:
6083:
6077:
6062:
6057:
6043:
6038:
6024:
6019:
6005:
6000:
5997:fivebrane group
5871:
5868:
5867:
5861:Whitehead tower
5857:
5855:Whitehead tower
5844:
5840:
5836:
5828:
5820:
5816:
5801:
5793:
5789:
5773:
5769:
5759:
5755:
5745:
5741:
5731:
5727:
5714:
5704:
5691:
5678:
5671:
5669:projective line
5660:
5655:
5647:
5639:
5633:
5627:
5618:
5599:
5595:
5575:
5571:
5551:
5547:
5531:
5527:
5523:In a nutshell:
5521:
5498:
5484:
5477:
5471:
5469:
5461:
5454:
5432:
5428:
5422:
5410:
5406:
5400:
5389:
5383:
5380:
5369:
5356:
5352:
5345:
5328:
5324:
5316:
5312:
5305:
5295:
5291:
5287:
5279:
5275:
5253:
5238:
5230:
5226:
5220:
5205:
5197:
5193:
5187:
5182:
5163:
5162:
5152:
5134:
5130:
5127:
5126:
5116:
5098:
5094:
5091:
5090:
5080:
5062:
5058:
5055:
5054:
5049:
5042:
5024:
5020:
5017:
5016:
5006:
4988:
4984:
4981:
4980:
4975:
4967:
4962:
4955:
4937:
4933:
4930:
4929:
4924:
4916:
4911:
4904:
4886:
4882:
4879:
4878:
4873:
4866:
4848:
4844:
4840:
4838:
4835:
4834:
4828:
4824:
4802:
4794:
4783:
4777:
4767:
4760:
4741:
4740:
4735:
4728:
4713:
4709:
4706:
4705:
4695:
4680:
4676:
4673:
4672:
4662:
4647:
4643:
4640:
4639:
4629:
4614:
4610:
4607:
4606:
4601:
4594:
4579:
4575:
4572:
4571:
4561:
4546:
4542:
4539:
4538:
4533:
4525:
4520:
4513:
4498:
4494:
4491:
4490:
4485:
4477:
4472:
4465:
4450:
4446:
4442:
4440:
4437:
4436:
4425:
4415:
4408:
4402:
4391:
4373:
4365:
4355:
4349:
4339:
4326:
4308:
4296:
4273:
4269:
4225:
4222:
4221:
4206:
4194:
4155:
4144:
4072:
4069:
4068:
4049:
4043:
4041:homotopy groups
4039:Generally, the
4037:
4035:Homotopy groups
4024:
4017:infinite cyclic
4009:
4006:universal cover
3997:
3978:
3967:
3960:
3938:
3930:
3920:
3914:
3905:
3901:
3896:
3880:
3870:
3861:
3850:
3844:
3841:
3833:help improve it
3830:
3821:
3817:
3806:
3795:
3789:
3786:
3775:
3762:
3758:
3751:
3738:
3734:
3716:
3715:
3710:
3704:
3703:
3698:
3688:
3687:
3685:
3682:
3681:
3673:
3661:
3653:
3645:
3641:
3632:
3615:
3611:
3602:
3598:
3584:
3580:
3565:
3561:
3560:
3556:
3554:
3551:
3550:
3546:
3538:
3529:
3521:
3504:
3500:
3491:
3487:
3469:
3465:
3450:
3446:
3444:
3441:
3440:
3432:
3425:
3421:
3397:
3396:
3391:
3385:
3384:
3379:
3369:
3368:
3362:
3354:
3353:
3348:
3342:
3341:
3336:
3326:
3325:
3323:
3320:
3319:
3312:
3308:
3304:
3299:
3291:
3284:
3280:
3276:
3271:
3260:
3256:
3252:
3249:symmetric group
3224:
3220:
3211:
3207:
3196:
3193:
3192:
3181:
3170:
3162:
3154:
3150:
3147:
3139:
3115:
3114:
3108:
3104:
3101:
3095:
3093:
3086:
3085:
3079:
3073:
3069:
3062:
3061:
3059:
3056:
3055:
3040:
3032:
3022:
3018:
3012:
2998:
2985:
2977:
2966:
2954:
2946:
2938:
2918:
2876:
2874:
2865:
2861:
2833:
2831:
2829:
2826:
2825:
2810:
2796:
2779:
2775:
2767:
2764:
2756:identity matrix
2743:
2731:
2720:
2702:
2694:
2691:
2689:Group structure
2674:
2670:
2658:
2634:
2633:
2616:
2598:
2597:
2577:
2555:
2554:
2552:
2549:
2548:
2541:
2514:
2511:complex numbers
2503:
2491:
2487:
2465:
2455:
2442:
2439:
2428:
2420:
2410:
2388:
2382:
2372:
2366:
2339:
2335:
2311:
2310:
2305:
2299:
2298:
2293:
2280:
2279:
2277:
2274:
2273:
2262:
2256:
2243:
2232:
2225:
2221:
2217:
2213:
2209:
2201:
2194:
2168:
2144:
2143:
2138:
2129:
2128:
2123:
2113:
2112:
2110:
2107:
2106:
2100:
2091:
2085:
2061:
2060:
2057:
2056:
2045:
2043:
2036:
2033:
2023:
2021:
2015:
2014:
2009:
2006:
2005:
1999:
1995:
1990:
1988:
1981:
1978:
1972:
1968:
1964:
1957:
1956:
1954:
1951:
1950:
1939:
1936:
1927:
1923:
1919:
1916:-fold rotations
1911:
1907:
1899:
1889:
1881:
1873:
1862:
1854:
1842:
1836:
1828:
1822:
1814:
1807:normal subgroup
1802:
1794:
1779:
1771:
1766:, as being the
1759:
1751:
1737:
1725:
1717:
1713:
1709:
1705:
1701:
1697:
1663:
1662:
1658:
1653:
1650:
1649:
1637:By choosing an
1635:
1622:
1614:
1599:
1587:
1584:normal subgroup
1577:
1558:
1477:
1474:
1473:
1462:
1454:
1448:
1434:
1420:
1409:
1399:
1385:
1379:
1377:Euclidean space
1364:
1316:
1313:
1312:
1306:
1286:
1275:
1274:The orthogonal
1272:
1257:
1251:
1240:uniform scaling
1227:
1221:
1215:
1209:
1135:
1134:
1130:
1114:
1113:
1109:
1078:
1074:
1048:
1045:
1044:
1032:
1017:
1007:
1001:
992:
982:
976:
969:
963:
957:
951:
931:
924:normal subgroup
909:
895:algebraic group
865:
847:
844:Euclidean space
827:
821:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
12763:
12753:
12752:
12747:
12742:
12737:
12723:
12722:
12714:
12709:
12704:
12684:
12683:External links
12681:
12679:
12678:
12657:
12644:
12640:978-3319134666
12639:
12626:
12613:
12592:
12579:
12571:Academic Press
12558:
12556:
12553:
12550:
12549:
12537:
12524:
12511:
12497:
12472:
12470:, p. 141)
12459:
12444:
12423:
12395:
12383:
12371:
12359:
12355:quadratic form
12347:characteristic
12337:
12336:
12334:
12331:
12328:
12327:
12298:
12266:
12252:
12251:
12249:
12246:
12245:
12244:
12242:Brauer algebra
12239:
12232:
12229:
12228:
12227:
12222:
12215:
12212:
12211:
12210:
12205:
12200:
12193:
12192:Related groups
12190:
12189:
12188:
12181:
12167:
12164:
12163:
12162:
12157:
12150:
12147:
12145:
12142:
12094:
12052:
12027:Main article:
12024:
12021:
11993:
11989:
11985:
11982:
11979:
11974:
11971:
11906:
11905:
11887:
11862:
11861:
11842:
11828:
11812:
11799:quotient group
11795:covering group
11782:
11779:
11778:
11777:
11766:
11755:
11748:
11694:
11691:
11626:} is the real
11598:direct product
11550:conformal maps
11534:Main article:
11531:
11528:
11527:
11526:
11511:
11508:
11505:
11501:
11497:
11494:
11492:
11488:
11483:
11478:
11477:
11474:
11471:
11468:
11464:
11460:
11457:
11455:
11451:
11446:
11441:
11440:
11437:
11434:
11431:
11427:
11423:
11420:
11418:
11414:
11409:
11404:
11403:
11400:
11397:
11394:
11390:
11386:
11383:
11381:
11377:
11372:
11367:
11366:
11363:
11360:
11357:
11353:
11349:
11346:
11344:
11340:
11335:
11330:
11329:
11326:
11323:
11320:
11316:
11312:
11309:
11307:
11305:
11302:
11299:
11296:
11292:
11289:
11286:
11282:
11281:
11278:
11275:
11272:
11268:
11264:
11261:
11259:
11257:
11254:
11251:
11247:
11243:
11242:
11219:
11216:
11215:
11214:
11201:
11196:
11191:
11188:
11185:
11182:
11178:
11167:
11156:
11153:
11150:
11146:
11143:
11140:
11136:
11133:
11130:
11127:
11124:
11120:
11109:
11073:
11070:
11067:
11063:
11060:
11056:
11053:
11050:
11047:
11043:
11039:
11036:
11033:
11030:
11027:
11023:
11012:
11000:
10997:
10994:
10991:
10988:
10984:
10980:
10977:
10974:
10971:
10967:
10947:
10944:
10880:
10879:Related groups
10877:
10845:
10842:
10839:
10836:
10833:
10828:
10825:
10812:
10811:
10800:
10797:
10794:
10791:
10788:
10785:
10782:
10779:
10776:
10773:
10770:
10767:
10764:
10761:
10758:
10755:
10752:
10722:
10719:
10716:
10713:
10710:
10707:
10704:
10701:
10698:
10695:
10692:
10689:
10686:
10683:
10680:
10660:
10583:
10563:
10527:
10524:
10521:
10518:
10515:
10510:
10507:
10485:
10482:
10479:
10476:
10473:
10468:
10442:skew-symmetric
10409:
10406:
10384:
10373:
10349:
10341:
10323:
10316:
10315:
10304:
10301:
10295:
10291:
10286:
10281:
10276:
10273:
10270:
10265:
10260:
10256:
10252:
10249:
10181:
10178:
10120:quotient group
10105:
10102:
10101:
10100:
10097:
10076:perfect fields
10064:
10047:
9937:
9934:
9834:characteristic
9779:
9776:
9752:
9751:
9740:
9736:
9732:
9729:
9724:
9721:
9717:
9712:
9706:
9703:
9700:
9695:
9692:
9689:
9685:
9680:
9676:
9673:
9668:
9664:
9659:
9653:
9650:
9647:
9644:
9641:
9638:
9634:
9630:
9627:
9623:
9619:
9616:
9613:
9610:
9607:
9604:
9601:
9596:
9592:
9587:
9576:
9565:
9561:
9557:
9554:
9549:
9546:
9542:
9537:
9531:
9528:
9525:
9520:
9517:
9514:
9510:
9505:
9501:
9498:
9493:
9489:
9484:
9478:
9475:
9472:
9469:
9466:
9463:
9459:
9455:
9452:
9448:
9444:
9441:
9438:
9435:
9432:
9429:
9426:
9421:
9417:
9412:
9401:
9390:
9386:
9382:
9379:
9374:
9371:
9367:
9362:
9356:
9351:
9348:
9345:
9341:
9333:
9329:
9324:
9320:
9317:
9313:
9309:
9306:
9303:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9278:
9251:
9245:
9242:
9235:
9232:
9229:
9225:
9221:
9216:
9213:
9209:
9202:
9199:
9196:
9193:
9171:
9165:
9162:
9159:
9155:
9151:
9146:
9143:
9139:
9132:
9129:
9126:
9123:
9112:
9111:
9096:
9091:
9085:
9082:
9079:
9076:
9074:
9071:
9068:
9065:
9062:
9061:
9058:
9055:
9052:
9049:
9047:
9044:
9041:
9038:
9037:
9035:
9030:
9027:
9025:
9023:
9020:
9019:
9016:
9012:
9008:
9005:
9002:
8999:
8996:
8993:
8990:
8988:
8985:
8981:
8980:
8962:
8961:
8946:
8941:
8938:
8934:
8930:
8927:
8925:
8923:
8920:
8919:
8916:
8913:
8910:
8908:
8905:
8901:
8898:
8894:
8889:
8885:
8884:
8859:
8853:
8850:
8843:
8840:
8836:
8832:
8829:
8823:
8820:
8798:
8792:
8789:
8785:
8781:
8778:
8772:
8769:
8758:
8757:
8742:
8737:
8731:
8728:
8726:
8723:
8720:
8719:
8716:
8713:
8711:
8708:
8707:
8705:
8700:
8697:
8695:
8693:
8690:
8689:
8686:
8681:
8676:
8671:
8668:
8665:
8662:
8657:
8653:
8649:
8646:
8644:
8641:
8637:
8636:
8618:
8603:
8592:
8591:
8576:
8571:
8568:
8565:
8562:
8559:
8556:
8552:
8548:
8545:
8543:
8541:
8538:
8537:
8534:
8531:
8528:
8526:
8523:
8519:
8516:
8513:
8510:
8507:
8503:
8498:
8494:
8493:
8462:
8434:roots of unity
8413:
8410:
8407:
8404:
8401:
8398:
8395:
8392:
8389:
8386:
8355:
8350:
8346:
8342:
8337:
8334:
8330:
8326:
8323:
8273:
8247:
8221:
8220:
8209:
8206:
8201:
8197:
8193:
8190:
8185:
8181:
8177:
8174:
8169:
8165:
8161:
8158:
8153:
8149:
8145:
8142:
8139:
8136:
8133:
8130:
8114:
8108:
8101:
8095:
8082:
8076:
8069:
8063:
8043:
8038:
8032:
8029:
8027:
8024:
8021:
8020:
8017:
8014:
8012:
8009:
8008:
8006:
7993:
7992:
7981:
7976:
7968:
7964:
7958:
7954:
7950:
7945:
7941:
7935:
7931:
7927:
7924:
7920:
7916:
7910:
7906:
7902:
7899:
7894:
7890:
7884:
7880:
7876:
7873:
7872:
7867:
7863:
7857:
7853:
7849:
7844:
7840:
7834:
7830:
7826:
7822:
7818:
7812:
7808:
7804:
7801:
7796:
7792:
7786:
7782:
7778:
7777:
7775:
7770:
7765:
7761:
7755:
7751:
7725:
7717:
7713:
7709:
7705:
7701:
7697:
7694:
7693:
7688:
7684:
7680:
7676:
7672:
7668:
7667:
7665:
7660:
7655:
7651:
7628:
7620:
7616:
7612:
7608:
7604:
7600:
7597:
7596:
7591:
7587:
7583:
7579:
7575:
7571:
7570:
7568:
7563:
7558:
7554:
7542:
7541:
7530:
7524:
7521:
7516:
7513:
7510:
7504:
7501:
7495:
7491:
7488:
7485:
7479:
7476:
7472:
7469:
7466:
7463:
7460:
7414:
7395:
7355:
7354:
7343:
7338:
7332:
7329:
7326:
7324:
7321:
7318:
7315:
7314:
7311:
7308:
7306:
7303:
7302:
7300:
7253:
7180:
7174:
7171:
7169:
7166:
7165:
7162:
7159:
7157:
7154:
7153:
7151:
7146:
7143:
7123:
7118:
7112:
7109:
7106:
7104:
7101:
7100:
7097:
7094:
7092:
7089:
7088:
7086:
7081:
7078:
7056:
7033:dihedral group
6970:
6965:
6959:
6956:
6954:
6951:
6950:
6947:
6944:
6942:
6939:
6938:
6936:
6834:
6829:
6823:
6820:
6819:
6817:
6791:
6785:
6782:
6781:
6779:
6696:
6690:
6687:
6685:
6682:
6681:
6678:
6675:
6673:
6670:
6669:
6667:
6651:
6636:
6631:
6630:
6619:
6616:
6613:
6608:
6604:
6600:
6597:
6594:
6589:
6585:
6581:
6576:
6572:
6568:
6565:
6523:
6520:
6518:
6515:
6330:
6323:
6312:quadratic form
6297:
6294:
6105:squares, with
6079:Main article:
6076:
6073:
6060:
6041:
6022:
6003:
5977:
5976:
5965:
5962:
5959:
5956:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5920:
5917:
5914:
5911:
5908:
5905:
5902:
5899:
5896:
5893:
5890:
5887:
5884:
5881:
5878:
5875:
5856:
5853:
5842:
5838:
5823:of the stable
5818:
5799:
5791:
5782:
5781:
5771:
5767:
5757:
5753:
5743:
5739:
5729:
5658:
5615:
5614:
5597:
5593:
5573:
5569:
5549:
5545:
5529:
5520:
5517:
5513:countably many
5467:
5453:
5450:
5430:
5408:
5402:
5401:
5359:
5357:
5350:
5344:
5343:Vector bundles
5341:
5326:
5314:
5304:
5301:
5300:
5299:
5293:
5285:
5277:
5273:
5236:
5228:
5224:
5203:
5195:
5186:
5183:
5181:
5178:
5177:
5176:
5161:
5158:
5155:
5153:
5151:
5148:
5145:
5142:
5137:
5133:
5129:
5128:
5125:
5122:
5119:
5117:
5115:
5112:
5109:
5106:
5101:
5097:
5093:
5092:
5089:
5086:
5083:
5081:
5079:
5076:
5073:
5070:
5065:
5061:
5057:
5056:
5052:
5048:
5045:
5043:
5041:
5038:
5035:
5032:
5027:
5023:
5019:
5018:
5015:
5012:
5009:
5007:
5005:
5002:
4999:
4996:
4991:
4987:
4983:
4982:
4978:
4974:
4970:
4965:
4961:
4958:
4956:
4954:
4951:
4948:
4945:
4940:
4936:
4932:
4931:
4927:
4923:
4919:
4914:
4910:
4907:
4905:
4903:
4900:
4897:
4894:
4889:
4885:
4881:
4880:
4876:
4872:
4869:
4867:
4865:
4862:
4859:
4856:
4851:
4847:
4843:
4842:
4826:
4789:
4779:
4759:
4756:
4755:
4754:
4738:
4734:
4731:
4729:
4727:
4724:
4721:
4716:
4712:
4708:
4707:
4704:
4701:
4698:
4696:
4694:
4691:
4688:
4683:
4679:
4675:
4674:
4671:
4668:
4665:
4663:
4661:
4658:
4655:
4650:
4646:
4642:
4641:
4638:
4635:
4632:
4630:
4628:
4625:
4622:
4617:
4613:
4609:
4608:
4604:
4600:
4597:
4595:
4593:
4590:
4587:
4582:
4578:
4574:
4573:
4570:
4567:
4564:
4562:
4560:
4557:
4554:
4549:
4545:
4541:
4540:
4536:
4532:
4528:
4523:
4519:
4516:
4514:
4512:
4509:
4506:
4501:
4497:
4493:
4492:
4488:
4484:
4480:
4475:
4471:
4468:
4466:
4464:
4461:
4458:
4453:
4449:
4445:
4444:
4421:
4410:
4361:
4351:
4293:
4292:
4281:
4276:
4272:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4187:
4186:
4175:
4172:
4169:
4166:
4163:
4158:
4153:
4150:
4147:
4143:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4045:
4036:
4033:
3959:
3956:
3955:
3954:
3942:(2) × SU(2) =
3935:doubly covered
3928:
3912:
3899:
3894:
3892:discrete space
3869:
3866:
3863:
3862:
3824:
3822:
3815:
3808:
3807:
3765:
3763:
3756:
3750:
3747:
3720:
3714:
3711:
3709:
3706:
3705:
3702:
3699:
3697:
3694:
3693:
3691:
3636:
3618:
3614:
3610:
3605:
3601:
3597:
3593:
3587:
3583:
3579:
3576:
3573:
3568:
3564:
3559:
3533:
3507:
3503:
3499:
3494:
3490:
3486:
3483:
3480:
3477:
3472:
3468:
3464:
3459:
3456:
3453:
3449:
3418:
3417:
3406:
3401:
3395:
3392:
3390:
3387:
3386:
3383:
3380:
3378:
3375:
3374:
3372:
3358:
3352:
3349:
3347:
3344:
3343:
3340:
3337:
3335:
3332:
3331:
3329:
3302:
3274:
3227:
3223:
3219:
3214:
3210:
3206:
3203:
3200:
3143:
3136:
3135:
3124:
3119:
3111:
3107:
3103:
3100:
3097:
3096:
3092:
3089:
3087:
3084:
3081:
3076:
3072:
3068:
3067:
3065:
2997:
2994:
2915:
2914:
2903:
2898:
2894:
2891:
2888:
2885:
2882:
2879:
2873:
2868:
2864:
2860:
2855:
2851:
2848:
2845:
2842:
2839:
2836:
2763:
2760:
2690:
2687:
2655:
2654:
2643:
2638:
2632:
2629:
2626:
2623:
2620:
2617:
2615:
2612:
2609:
2606:
2603:
2600:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2576:
2573:
2570:
2567:
2564:
2561:
2560:
2558:
2538:absolute value
2480:symmetry group
2451:symmetry group
2438:
2435:
2332:
2331:
2320:
2315:
2309:
2306:
2304:
2301:
2300:
2297:
2294:
2292:
2289:
2286:
2285:
2283:
2255:
2252:
2185:by regrouping
2165:
2164:
2153:
2148:
2142:
2139:
2137:
2134:
2131:
2130:
2127:
2124:
2122:
2119:
2118:
2116:
2096:
2089:
2082:
2081:
2070:
2065:
2055:
2052:
2049:
2046:
2042:
2039:
2037:
2032:
2029:
2026:
2025:
2022:
2020:
2017:
2016:
2013:
2010:
2002:
1998:
1994:
1991:
1987:
1984:
1982:
1975:
1971:
1967:
1966:
1963:
1962:
1960:
1935:
1934:Canonical form
1932:
1903:
1851:direct product
1748:of the space.
1690:
1689:
1678:
1675:
1672:
1666:
1661:
1657:
1634:
1631:
1550:
1549:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1303:Euclidean norm
1271:
1268:
1208:
1205:
1174:quadratic form
1155:
1151:
1147:
1144:
1138:
1133:
1129:
1126:
1123:
1117:
1112:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1077:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
944:rotation group
930:, and denoted
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
9:
6:
4:
3:
2:
12762:
12751:
12748:
12746:
12743:
12741:
12738:
12736:
12733:
12732:
12730:
12721:
12715:
12713:
12710:
12708:
12705:
12701:
12697:
12696:
12691:
12687:
12686:
12676:
12672:
12668:
12664:
12660:
12658:3-88538-009-9
12654:
12650:
12645:
12642:
12636:
12632:
12627:
12624:
12620:
12616:
12610:
12606:
12602:
12598:
12593:
12590:
12586:
12582:
12580:0-12-163260-1
12576:
12572:
12568:
12564:
12560:
12559:
12547:, p. 178
12546:
12541:
12534:
12528:
12521:
12515:
12508:
12504:
12500:
12498:3-540-52117-8
12494:
12490:
12486:
12479:
12477:
12469:
12463:
12455:
12451:
12447:
12441:
12437:
12430:
12428:
12413:
12409:
12405:
12399:
12392:
12387:
12381:Section 1.3.4
12380:
12375:
12368:
12363:
12356:
12352:
12348:
12342:
12338:
12322:
12314:
12310:
12302:
12289:
12285:
12281:
12277:
12270:
12263:
12257:
12253:
12243:
12240:
12238:
12235:
12234:
12226:
12223:
12221:
12218:
12217:
12209:
12206:
12204:
12203:unitary group
12201:
12199:
12196:
12195:
12186:
12182:
12180:
12177:
12170:
12169:
12161:
12158:
12156:
12153:
12152:
12141:
12138:
12132:
12127:
12122:
12117:
12112:
12108:
12101:
12097:
12088:
12086:
12082:
12076:
12074:
12071:
12066:(orthonormal
12065:
12059:
12055:
12048:
12042:
12036:
12030:
12020:
12017:
11983:
11980:
11959:
11953:
11945:
11937:
11929:
11921:
11913:
11901:
11897:
11891:
11888:
11883:
11879:
11873:
11870:
11869:
11868:
11865:
11857:
11853:
11847:
11844:The quotient
11843:
11838:
11834:
11822:
11818:
11808:
11805:Two covering
11804:
11803:
11802:
11800:
11796:
11792:
11788:
11775:
11769:
11763:
11762:Coxeter group
11759:
11756:
11751:
11745:
11744:Coxeter group
11741:
11738:
11737:
11736:
11733:
11731:
11727:
11723:
11719:
11714:
11712:
11708:
11704:
11700:
11690:
11687:
11683:
11679:
11675:
11671:
11663:
11656:
11653:
11649:
11645:
11634:
11629:
11623:
11619:
11613:
11609:
11605:
11599:
11594:
11589:
11585:
11579:
11572:
11567:
11563:
11559:
11555:
11551:
11547:
11543:
11537:
11506:
11495:
11493:
11486:
11469:
11458:
11456:
11449:
11432:
11421:
11419:
11412:
11395:
11384:
11382:
11375:
11358:
11347:
11345:
11338:
11321:
11310:
11308:
11300:
11297:
11273:
11262:
11260:
11252:
11233:
11232:
11231:
11227:
11199:
11189:
11183:
11168:
11151:
11134:
11128:
11125:
11110:
11105:
11099:
11095:
11089:
11068:
11054:
11048:
11037:
11031:
11028:
11013:
10995:
10992:
10989:
10978:
10972:
10957:
10956:
10955:
10953:
10946:Lie subgroups
10943:
10941:
10935:
10931:
10925:
10921:
10915:
10911:
10907:
10899:
10895:
10891:
10884:
10876:
10874:
10869:
10865:
10861:
10840:
10837:
10834:
10798:
10792:
10789:
10786:
10780:
10777:
10771:
10768:
10765:
10756:
10753:
10750:
10743:
10742:
10741:
10738:
10720:
10717:
10711:
10708:
10705:
10702:
10696:
10690:
10687:
10684:
10681:
10658:
10645:
10639:
10637:
10633:
10629:
10623:
10617:
10613:
10607:
10600:
10597:
10593:
10586:
10577:
10573:
10566:
10559:
10555:
10550:
10545:
10541:
10522:
10519:
10516:
10480:
10477:
10474:
10456:
10451:
10447:
10443:
10437:
10433:
10425:
10421:
10415:
10405:
10402:
10396:
10390:
10382:
10372:
10368:
10362:
10355:
10339:
10334:
10330:
10322:
10302:
10279:
10258:
10254:
10247:
10240:
10239:
10238:
10236:
10232:
10228:
10224:
10220:
10215:
10213:
10209:
10204:
10199:
10195:
10191:
10187:
10177:
10173:
10169:
10165:
10159:
10153:
10148:
10144:
10140:
10136:of the field
10135:
10129:
10125:
10121:
10116:
10111:
10098:
10094:
10087:
10082:in dimension
10081:
10077:
10071:
10065:
10061:
10057:
10052:
10048:
10044:
10040:
10036:
10032:
10028:
10024:
10020:
10014:
10009:
10004:
9998:
9992:
9988:
9984:
9980:
9976:
9972:
9968:
9962:
9956:
9951:
9947:
9946:
9945:
9943:
9933:
9931:
9927:
9922:
9910:
9906:
9894:
9890:
9882:
9878:
9863:
9856:
9852:
9846:
9841:
9835:
9831:
9826:
9819:
9815:
9811:
9807:
9801:
9794:
9790:
9785:
9775:
9769:
9765:
9738:
9734:
9730:
9727:
9722:
9719:
9715:
9710:
9704:
9701:
9698:
9693:
9690:
9687:
9683:
9678:
9674:
9671:
9666:
9662:
9657:
9648:
9645:
9642:
9636:
9632:
9628:
9625:
9621:
9614:
9611:
9608:
9605:
9599:
9594:
9585:
9577:
9563:
9559:
9555:
9552:
9547:
9544:
9540:
9535:
9529:
9526:
9523:
9518:
9515:
9512:
9508:
9503:
9499:
9496:
9491:
9487:
9482:
9473:
9470:
9467:
9461:
9457:
9453:
9450:
9446:
9439:
9436:
9433:
9430:
9424:
9419:
9410:
9402:
9388:
9384:
9380:
9377:
9372:
9369:
9365:
9360:
9354:
9349:
9346:
9343:
9339:
9331:
9327:
9322:
9318:
9315:
9311:
9304:
9301:
9298:
9295:
9292:
9289:
9283:
9276:
9268:
9267:
9266:
9262:
9249:
9243:
9240:
9233:
9230:
9227:
9223:
9219:
9214:
9211:
9207:
9200:
9197:
9194:
9191:
9169:
9163:
9160:
9157:
9153:
9149:
9144:
9141:
9137:
9130:
9127:
9124:
9121:
9094:
9089:
9083:
9080:
9077:
9072:
9069:
9066:
9063:
9056:
9053:
9050:
9045:
9042:
9039:
9033:
9026:
9021:
9006:
9003:
8997:
8994:
8989:
8971:
8970:
8969:
8944:
8939:
8936:
8932:
8926:
8921:
8914:
8909:
8899:
8896:
8892:
8875:
8874:
8873:
8870:
8857:
8851:
8848:
8841:
8838:
8834:
8830:
8827:
8821:
8818:
8796:
8790:
8787:
8783:
8779:
8776:
8770:
8767:
8740:
8735:
8729:
8724:
8721:
8714:
8709:
8703:
8696:
8691:
8679:
8669:
8666:
8660:
8655:
8651:
8645:
8627:
8626:
8625:
8621:
8617:
8606:
8602:
8574:
8569:
8563:
8560:
8557:
8550:
8544:
8539:
8532:
8527:
8514:
8511:
8508:
8501:
8484:
8483:
8482:
8479:
8476:
8469:
8465:
8461:
8456:
8447:
8440:
8435:
8429:
8411:
8408:
8405:
8402:
8393:
8390:
8387:
8370:
8353:
8348:
8344:
8340:
8335:
8332:
8328:
8324:
8321:
8313:
8309:
8295:
8291:
8284:
8276:
8272:
8265:
8261:
8254:
8250:
8246:
8240:
8236:
8232:
8228:
8207:
8199:
8195:
8191:
8188:
8183:
8179:
8167:
8163:
8159:
8156:
8151:
8147:
8140:
8137:
8134:
8131:
8128:
8121:
8120:
8119:
8113:
8107:
8100:
8094:
8090:
8081:
8075:
8068:
8062:
8058:
8041:
8036:
8030:
8025:
8022:
8015:
8010:
8004:
7979:
7974:
7966:
7962:
7956:
7952:
7948:
7943:
7939:
7933:
7929:
7925:
7918:
7914:
7908:
7904:
7900:
7897:
7892:
7888:
7882:
7878:
7874:
7865:
7861:
7855:
7851:
7847:
7842:
7838:
7832:
7828:
7820:
7816:
7810:
7806:
7802:
7799:
7794:
7790:
7784:
7780:
7773:
7768:
7763:
7759:
7753:
7749:
7741:
7740:
7739:
7723:
7715:
7711:
7703:
7699:
7695:
7686:
7682:
7674:
7670:
7663:
7658:
7653:
7649:
7626:
7618:
7614:
7606:
7602:
7598:
7589:
7585:
7577:
7573:
7566:
7561:
7556:
7552:
7528:
7522:
7519:
7514:
7511:
7508:
7502:
7499:
7493:
7489:
7486:
7483:
7477:
7474:
7470:
7467:
7464:
7461:
7458:
7451:
7450:
7449:
7446:
7442:
7438:
7432:
7428:
7424:
7417:
7413:
7406:
7398:
7394:
7380:
7372:
7365:
7361:
7341:
7336:
7330:
7327:
7322:
7319:
7316:
7309:
7304:
7298:
7289:
7288:
7287:
7283:
7277:
7273:
7267:
7263:
7256:
7252:
7234:
7230:
7226:
7219:
7215:
7208:
7204:
7197:
7178:
7172:
7167:
7160:
7155:
7149:
7144:
7141:
7121:
7116:
7110:
7107:
7102:
7095:
7090:
7084:
7079:
7076:
7066:
7055:
7051:
7044:
7040:
7034:
7028:
7021:
7017:
7013:
7005:
7001:
6986:
6968:
6963:
6957:
6952:
6945:
6940:
6934:
6898:
6867:
6859:
6855:
6832:
6827:
6821:
6815:
6789:
6783:
6777:
6757:
6751:
6747:
6742:
6730:
6726:
6721:
6694:
6688:
6683:
6676:
6671:
6665:
6655:has the form
6643:
6617:
6614:
6611:
6606:
6602:
6598:
6595:
6592:
6587:
6583:
6579:
6574:
6570:
6566:
6563:
6556:
6555:
6554:
6548:
6543:
6539:
6537:
6533:
6529:
6514:
6511:
6504:
6498:
6492:
6488:
6482:
6476:
6469:
6465:
6457:
6451:
6445:
6438:
6434:
6427:
6423:
6415:
6411:
6404:
6400:
6396:
6388:
6384:
6372:
6368:
6361:
6359:
6353:
6349:
6340:of dimension
6339:
6333:
6329:
6322:
6313:
6309:
6304:
6293:
6283:
6279:
6278:Lorentz group
6270:
6266:
6262:
6253:
6249:
6241:
6237:
6229:
6225:
6218:
6206:
6202:
6198:
6191:
6187:
6183:
6179:
6175:
6167:
6163:
6156:
6154:
6148:
6144:
6122:
6117:
6113:
6109:
6092:
6088:
6082:
6072:
6070:
6066:
6055:
6051:
6047:
6036:
6032:
6028:
6017:
6013:
6009:
5998:
5994:
5990:
5986:
5982:
5960:
5954:
5942:
5936:
5933:
5924:
5918:
5915:
5906:
5900:
5897:
5888:
5882:
5879:
5873:
5866:
5865:
5864:
5862:
5852:
5848:
5832:
5826:
5814:
5809:
5805:
5797:
5787:
5777:
5768:
5763:
5754:
5749:
5740:
5735:
5726:
5725:
5724:
5721:
5717:
5711:
5707:
5701:
5697:
5695:
5688:
5684:
5682:
5674:
5670:
5666:
5661:
5652:
5651:
5644:
5643:
5636:
5630:
5626:
5621:
5612:
5607:
5603:
5594:
5592:
5587:
5583:
5579:
5570:
5568:
5563:
5559:
5555:
5546:
5544:
5539:
5535:
5526:
5525:
5524:
5516:
5514:
5509:
5505:
5501:
5495:
5491:
5487:
5480:
5474:
5464:
5459:
5449:
5447:
5442:
5441:
5436:
5425:
5420:
5419:vector bundle
5414:
5398:
5395:
5387:
5377:
5376:the talk page
5373:
5367:
5365:
5360:This section
5358:
5349:
5348:
5340:
5338:
5332:
5320:
5310:
5283:
5274:
5269:
5265:
5261:
5257:
5251:
5246:
5242:
5234:
5225:
5218:
5213:
5209:
5201:
5192:
5191:
5190:
5159:
5156:
5154:
5146:
5143:
5135:
5131:
5123:
5120:
5118:
5110:
5107:
5099:
5095:
5087:
5084:
5082:
5074:
5071:
5063:
5059:
5046:
5044:
5036:
5033:
5025:
5021:
5013:
5010:
5008:
5000:
4997:
4989:
4985:
4972:
4968:
4959:
4957:
4949:
4946:
4938:
4934:
4921:
4917:
4908:
4906:
4898:
4895:
4887:
4883:
4870:
4868:
4860:
4857:
4849:
4845:
4833:
4832:
4831:
4821:
4817:
4813:
4809:
4805:
4798:
4792:
4787:
4782:
4775:
4770:
4765:
4732:
4730:
4722:
4714:
4710:
4702:
4699:
4697:
4689:
4681:
4677:
4669:
4666:
4664:
4656:
4648:
4644:
4636:
4633:
4631:
4623:
4615:
4611:
4598:
4596:
4588:
4580:
4576:
4568:
4565:
4563:
4555:
4547:
4543:
4530:
4526:
4517:
4515:
4507:
4499:
4495:
4482:
4478:
4469:
4467:
4459:
4451:
4447:
4435:
4434:
4433:
4429:
4424:
4419:
4413:
4405:
4399:
4395:
4389:
4384:
4380:
4376:
4369:
4364:
4359:
4354:
4347:
4343:
4334:
4330:
4324:
4320:
4316:
4311:
4306:
4300:
4279:
4274:
4270:
4260:
4257:
4254:
4248:
4236:
4230:
4220:
4219:
4218:
4216:
4210:
4204:
4199:
4198:
4192:
4170:
4164:
4151:
4148:
4145:
4141:
4137:
4134:
4131:
4128:
4125:
4119:
4113:
4107:
4101:
4095:
4089:
4083:
4077:
4067:
4066:
4065:
4063:
4059:
4053:
4048:
4042:
4032:
4030:
4022:
4018:
4012:
4007:
4001:
3996:
3992:
3986:
3982:
3976:
3970:
3965:
3951:
3947:
3946:
3941:
3936:
3929:
3926:
3924:
3917:
3913:
3910:
3909:
3900:
3895:
3893:
3889:
3884:
3879:
3878:
3877:
3875:
3859:
3856:
3848:
3845:November 2019
3838:
3834:
3828:
3825:This section
3823:
3814:
3813:
3804:
3801:
3793:
3790:November 2019
3783:
3782:the talk page
3779:
3773:
3771:
3766:This section
3764:
3755:
3754:
3746:
3742:
3718:
3712:
3707:
3700:
3695:
3689:
3677:
3669:
3665:
3657:
3649:
3639:
3635:
3616:
3612:
3608:
3603:
3599:
3591:
3585:
3581:
3577:
3574:
3571:
3566:
3562:
3557:
3544:
3536:
3532:
3525:
3505:
3501:
3497:
3492:
3484:
3481:
3475:
3470:
3466:
3462:
3457:
3454:
3451:
3447:
3436:
3429:
3404:
3399:
3393:
3388:
3381:
3376:
3370:
3356:
3350:
3345:
3338:
3333:
3327:
3318:
3317:
3316:
3305:
3295:
3287:
3279:acts on both
3277:
3269:
3263:
3250:
3246:
3243:
3225:
3221:
3217:
3212:
3204:
3201:
3191:
3185:
3179:
3174:
3166:
3158:
3146:
3142:
3122:
3117:
3109:
3105:
3098:
3090:
3082:
3074:
3070:
3063:
3054:
3053:
3052:
3051:of the form
3050:
3044:
3036:
3029:
3025:
3015:
3010:
3007:
3004:in a compact
3003:
3002:maximal torus
2993:
2989:
2981:
2973:
2969:
2964:
2958:
2950:
2942:
2936:
2932:
2928:
2922:
2901:
2896:
2889:
2886:
2883:
2877:
2871:
2866:
2862:
2858:
2853:
2846:
2843:
2840:
2834:
2824:
2823:
2822:
2820:
2819:algebraic set
2814:
2807:
2803:
2799:
2794:
2789:
2785:
2782:
2771:
2759:
2757:
2753:
2747:
2741:
2735:
2727:
2723:
2719:
2715:
2712:
2706:
2698:
2686:
2684:
2678:
2668:
2662:
2641:
2636:
2627:
2621:
2618:
2610:
2604:
2601:
2591:
2585:
2582:
2579:
2571:
2565:
2562:
2556:
2547:
2546:
2545:
2539:
2533:
2529:
2525:
2521:
2518:
2512:
2506:
2501:
2497:
2485:
2481:
2476:
2474:
2468:
2463:
2459:
2452:
2446:
2434:
2433:reflections.
2431:
2424:
2417:
2413:
2408:
2403:
2401:
2396:
2392:
2385:
2379:
2375:
2369:
2364:
2359:
2357:
2353:
2347:
2343:
2318:
2313:
2307:
2302:
2295:
2290:
2287:
2281:
2272:
2271:
2270:
2266:
2260:
2251:
2249:
2241:
2235:
2229:
2205:
2198:
2192:
2188:
2184:
2179:
2175:
2171:
2151:
2146:
2140:
2135:
2132:
2125:
2120:
2114:
2105:
2104:
2103:
2099:
2095:
2088:
2068:
2063:
2053:
2050:
2040:
2030:
2027:
2018:
2011:
2000:
1996:
1985:
1973:
1969:
1958:
1949:
1948:
1947:
1943:
1931:
1917:
1914:
1906:
1902:
1898:
1892:
1885:
1879:
1870:
1866:
1858:
1852:
1846:
1839:
1832:
1825:
1818:
1812:
1808:
1798:
1791:
1789:
1783:
1777:
1769:
1763:
1755:
1749:
1747:
1741:
1735:
1729:
1723:
1695:
1676:
1673:
1670:
1659:
1655:
1648:
1647:
1646:
1644:
1640:
1630:
1626:
1618:
1612:
1607:
1603:
1597:
1591:
1585:
1580:
1575:
1570:
1568:
1567:
1561:
1555:
1536:
1530:
1524:
1521:
1515:
1509:
1506:
1500:
1497:
1494:
1485:
1479:
1472:
1471:
1470:
1466:
1458:
1451:
1447:
1442:
1438:
1431:
1427:
1423:
1416:
1412:
1406:
1402:
1397:
1393:
1388:
1384:of dimension
1382:
1378:
1374:
1368:
1361:
1348:
1342:
1336:
1327:
1321:
1309:
1304:
1300:
1299:endomorphisms
1294:
1290:
1285:
1279:
1267:
1265:
1260:
1254:
1249:
1245:
1241:
1237:
1231:
1224:
1220:of dimension
1218:
1214:
1204:
1202:
1197:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1166:
1153:
1149:
1145:
1142:
1131:
1127:
1124:
1121:
1110:
1106:
1100:
1097:
1094:
1088:
1085:
1082:
1079:
1075:
1071:
1065:
1062:
1059:
1053:
1040:
1036:
1031:
1025:
1021:
1014:
1010:
1004:
1000:
995:
989:
985:
979:
973:
966:
960:
954:
949:
945:
941:
935:
929:
926:, called the
925:
921:
917:
912:
906:
904:
900:
896:
892:
888:
884:
880:
876:
872:
868:
863:
859:
855:
850:
846:of dimension
845:
841:
837:
831:
824:
820:in dimension
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
19:
12717:(in Italian)
12693:
12648:
12630:
12596:
12566:
12545:Cassels 1978
12540:
12527:
12514:
12484:
12462:
12435:
12415:. Retrieved
12411:
12398:
12386:
12374:
12369:Theorem 11.2
12362:
12341:
12320:
12312:
12311:+ 1) ≅ PSO(2
12308:
12301:
12287:
12283:
12275:
12269:
12256:
12175:
12136:
12130:
12125:
12120:
12115:
12110:
12106:
12099:
12092:
12089:
12084:
12080:
12077:
12069:
12057:
12050:
12040:
12032:
12018:
11951:
11943:
11935:
11927:
11919:
11911:
11907:
11899:
11895:
11881:
11877:
11866:
11863:
11855:
11851:
11836:
11832:
11820:
11816:
11784:
11767:
11749:
11734:
11715:
11699:point groups
11696:
11685:
11681:
11677:
11673:
11669:
11661:
11657:
11651:
11647:
11643:
11632:
11621:
11617:
11611:
11607:
11603:
11592:
11583:
11577:
11570:
11539:
11225:
11221:
11103:
11093:
11087:
10952:Kaluza–Klein
10949:
10933:
10929:
10913:
10909:
10905:
10897:
10893:
10889:
10885:
10882:
10870:
10863:
10859:
10813:
10736:
10643:
10640:
10631:
10621:
10615:
10611:
10605:
10601:
10595:
10591:
10584:
10575:
10571:
10564:
10548:
10543:
10539:
10449:
10445:
10435:
10431:
10423:
10419:
10411:
10400:
10394:
10391:
10380:
10370:
10366:
10360:
10353:
10337:
10320:
10317:
10216:
10202:
10193:
10183:
10174:
10167:
10163:
10157:
10151:
10138:
10127:
10123:
10114:
10109:
10107:
10092:
10085:
10069:
10059:
10055:
10042:
10038:
10034:
10030:
10026:
10022:
10018:
10012:
10002:
9996:
9990:
9986:
9982:
9978:
9974:
9970:
9966:
9960:
9954:
9941:
9939:
9923:
9908:
9904:
9892:
9888:
9880:
9876:
9861:
9854:
9850:
9842:
9824:
9817:
9813:
9809:
9805:
9802:
9792:
9788:
9783:
9781:
9767:
9763:
9753:
9264:
9113:
8963:
8871:
8759:
8619:
8615:
8604:
8600:
8593:
8480:
8477:
8470:
8463:
8459:
8445:
8438:
8427:
8368:
8298:, the above
8293:
8289:
8285:
8274:
8270:
8263:
8259:
8255:
8248:
8244:
8238:
8234:
8230:
8226:
8222:
8111:
8105:
8098:
8092:
8088:
8079:
8073:
8066:
8060:
8056:
7994:
7543:
7444:
7440:
7436:
7430:
7426:
7422:
7415:
7411:
7404:
7396:
7392:
7381:
7370:
7363:
7359:
7356:
7281:
7275:
7271:
7265:
7261:
7259:, such that
7254:
7250:
7232:
7228:
7224:
7217:
7213:
7206:
7202:
7195:
7064:
7060:
7049:
7042:
7038:
7026:
7022:
7015:
7011:
7003:
6999:
6987:
6896:
6868:
6857:
6853:
6758:
6750:finite field
6743:
6728:
6724:
6632:
6544:
6540:
6531:
6525:
6509:
6502:
6490:
6486:
6474:
6467:
6463:
6455:
6449:
6443:
6436:
6432:
6425:
6421:
6413:
6409:
6405:
6398:
6394:
6386:
6382:
6370:
6366:
6362:
6357:
6351:
6347:
6331:
6327:
6320:
6302:
6299:
6271:
6264:
6260:
6251:
6247:
6244:. The group
6239:
6235:
6227:
6223:
6219:
6204:
6203:, 0) = O(0,
6200:
6196:
6192:
6185:
6181:
6177:
6173:
6165:
6161:
6157:
6152:
6146:
6142:
6115:
6111:
6107:
6084:
6064:
6053:
6049:
6048:) to obtain
6045:
6034:
6030:
6029:) to obtain
6026:
6015:
6011:
6010:) to obtain
6007:
5993:string group
5978:
5858:
5846:
5830:
5813:Maslov index
5807:
5803:
5795:
5783:
5775:
5761:
5747:
5733:
5719:
5715:
5709:
5705:
5699:
5693:
5686:
5680:
5672:
5656:
5648:
5640:
5634:
5628:
5619:
5616:
5605:
5601:
5585:
5581:
5577:
5561:
5557:
5553:
5537:
5533:
5522:
5507:
5503:
5499:
5493:
5489:
5485:
5478:
5472:
5462:
5455:
5438:
5434:
5423:
5412:
5405:
5390:
5384:January 2024
5381:
5370:Please help
5361:
5337:free abelian
5330:
5318:
5306:
5281:
5267:
5263:
5259:
5255:
5244:
5240:
5232:
5211:
5207:
5199:
5188:
4819:
4815:
4811:
4807:
4803:
4796:
4790:
4785:
4780:
4768:
4761:
4427:
4422:
4417:
4411:
4403:
4397:
4393:
4385:
4378:
4374:
4367:
4362:
4357:
4352:
4341:
4332:
4328:
4309:
4305:transitively
4298:
4294:
4215:fiber bundle
4208:
4196:
4191:cofibrations
4188:
4062:direct limit
4051:
4046:
4038:
4029:2-fold cover
4010:
3999:
3984:
3980:
3968:
3962:In terms of
3961:
3949:
3944:
3922:
3907:
3882:
3871:
3851:
3842:
3826:
3796:
3787:
3776:Please help
3767:
3740:
3675:
3667:
3663:
3655:
3647:
3637:
3633:
3534:
3530:
3523:
3434:
3430:
3419:
3300:
3293:
3285:
3272:
3261:
3240:of a normal
3183:
3175:
3164:
3156:
3144:
3140:
3137:
3042:
3034:
3030:
3023:
3013:
3008:
2999:
2987:
2979:
2971:
2967:
2961:). Both are
2956:
2948:
2940:
2920:
2916:
2812:
2808:
2801:
2797:
2787:
2783:
2780:
2769:
2765:
2745:
2733:
2730:. The group
2725:
2721:
2704:
2696:
2692:
2676:
2673:-sphere and
2660:
2656:
2531:
2527:
2523:
2519:
2516:
2500:circle group
2495:
2477:
2466:
2457:
2444:
2440:
2429:
2422:
2415:
2411:
2404:
2397:
2390:
2383:
2380:
2373:
2367:
2360:
2352:mirror image
2345:
2341:
2333:
2264:
2257:
2238:is known as
2233:
2230:
2203:
2199:
2180:
2173:
2169:
2166:
2097:
2093:
2086:
2083:
1941:
1937:
1912:
1904:
1900:
1897:cyclic group
1890:
1883:
1871:
1864:
1856:
1844:
1837:
1830:
1823:
1816:
1796:
1792:
1781:
1761:
1753:
1750:
1739:
1727:
1721:
1691:
1636:
1624:
1616:
1608:
1601:
1589:
1578:
1571:
1564:
1559:
1551:
1464:
1456:
1449:
1443:
1436:
1429:
1425:
1421:
1414:
1410:
1404:
1400:
1386:
1380:
1366:
1362:
1307:
1292:
1288:
1277:
1273:
1258:
1252:
1229:
1222:
1216:
1210:
1198:
1185:
1178:vector space
1167:
1038:
1034:
1023:
1019:
1012:
1008:
1002:
998:
993:
987:
983:
977:
974:
943:
933:
927:
910:
907:
870:
866:
857:
848:
829:
822:
817:
811:
640:
628:
616:
604:
592:
580:
576:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
12522:, page 160)
12520:Taylor 1992
12468:Taylor 1992
11630:, while if
11574:is denoted
10414:Lie algebra
10408:Lie algebra
10212:determinant
10110:spinor norm
9830:Taylor 1992
7200:, that is,
6908:is two and
6873:is two and
6720:anisotropic
6633:where each
6406:The groups
6284:. Here the
6232:is denoted
6151:called the
6139:. The pair
6056:, and then
5567:orientation
5452:Loop spaces
5288:(SO(3)) = 0
5252:comes from
5248:, which is
5217:orientation
4319:unit vector
4023:(the group
3897:SO(1) = {1}
3631:; that is,
3547:{±1} → {±1}
3520:of that of
3149:belongs to
3138:where each
2937:. In fact,
2693:The groups
2667:topological
2259:Reflections
2254:Reflections
2187:eigenvalues
1809:and even a
1746:orientation
1694:determinant
1554:translation
1398:of a point
1248:linear maps
1194:dot product
1190:linear maps
1043:; that is
940:determinant
889:equals its
883:real matrix
814:mathematics
530:Topological
369:alternating
12735:Lie groups
12729:Categories
12675:0767.20001
12589:0395.10029
12555:References
12533:Grove 2002
12507:0756.11008
12454:1203.20012
12417:2023-02-01
12408:"Week 105"
12404:Baez, John
12116:incomplete
12085:orthogonal
12081:orthogonal
11872:Spin group
11807:Pin groups
11791:centerless
11606:+ 1) = O(2
11558:similarity
11554:congruence
11542:isometries
10628:spin group
10616:projective
10455:commutator
10219:spin group
9950:Witt index
9930:pin groups
9820:) modulo 2
7448:, one has
6722:(that is,
6532:equivalent
6272:The group
5989:spin group
5654:, and let
5366:to readers
5309:Lie groups
5303:Lie groups
5239:(SO(3)) =
4801:. Setting
4390:we obtain
4346:-connected
4315:stabilizer
4313:, and the
3995:spin group
3993:, and the
3772:to readers
3255:factor of
3245:2-subgroup
3178:Weyl group
2778:such that
2750:being the
2714:Lie groups
2356:hyperplane
1872:The group
1821:, and, if
1788:reflection
1724:, denoted
1708:is either
1419:such that
1392:isomorphic
1311:such that
826:, denoted
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
12700:EMS Press
12391:Hall 2015
12379:Hall 2015
12367:Hall 2015
12333:Citations
12126:principal
11703:polytopes
11588:dilations
11496:⊃
11459:⊃
11422:⊃
11385:⊃
11348:⊃
11311:⊃
11263:⊃
11190:⊃
11135:⊃
11055:⊃
11038:⊃
10993:−
10979:⊃
10896:) ⊂ USp(2
10793:⋅
10781:−
10772:⋅
10760:↦
10754:∧
10709:φ
10682:φ
10659:φ
10300:→
10285:→
10264:→
10255:μ
10251:→
10223:pin group
9812:) = rank(
9728:−
9702:−
9684:∏
9646:−
9600:
9595:−
9553:−
9527:−
9509:∏
9497:−
9471:−
9425:
9378:−
9340:∏
9284:
9234:θ
9228:−
9220:−
9215:θ
9198:θ
9195:
9164:θ
9158:−
9145:θ
9128:θ
9125:
9084:θ
9081:
9073:θ
9070:
9064:−
9057:θ
9054:
9046:θ
9043:
9029:↦
8998:
8992:→
8940:θ
8929:↦
8922:θ
8912:→
8900:π
8852:α
8839:−
8831:−
8788:−
8722:ω
8699:↦
8661:
8648:→
8561:−
8547:↦
8530:→
8409:α
8400:↦
8366:and thus
8333:−
8308:conjugate
8192:α
8160:α
8135:α
8023:ω
7926:ω
7901:ω
7875:ω
7803:ω
7696:ω
7599:ω
7523:α
7512:−
7409:, and to
7328:ε
7320:ω
7317:ε
7111:ω
7108:−
7035:of order
6958:φ
6822:φ
6612:⊕
6599:⊕
6596:⋯
6593:⊕
6580:⊕
6536:congruent
6217:is zero.
5955:
5949:→
5937:
5931:→
5919:
5913:→
5901:
5895:→
5883:
5880:Fivebrane
5877:→
5874:⋯
5841:(U/O) = π
5667:over the
5609:is about
5589:is about
5565:is about
5543:dimension
5541:is about
5446:dimension
5335:is free (
5206:(O(1)) =
5132:π
5096:π
5060:π
5022:π
4986:π
4935:π
4884:π
4846:π
4823:(to make
4711:π
4678:π
4645:π
4612:π
4577:π
4544:π
4496:π
4448:π
4360:+ 1)) = π
4267:→
4249:
4243:→
4231:
4165:
4157:∞
4142:⋃
4132:⊂
4129:⋯
4126:⊂
4114:
4108:⊂
4096:
4090:⊂
4078:
4021:real line
3642:< {±1}
3613:ε
3609:⋯
3600:ε
3596:↦
3582:ε
3575:…
3563:ε
3549:given by
3539:< {±1}
3498:⋊
3482:±
3463:⋊
3455:−
3268:inversion
3218:⋊
3202:±
3091:⋱
3017:for some
3006:Lie group
2872:−
2844:−
2718:dimension
2709:are real
2628:φ
2622:
2611:φ
2605:
2592:φ
2586:
2580:−
2572:φ
2566:
2409:(the map
2288:−
2189:that are
2133:−
2051:±
2041:⋱
2028:±
1986:⋱
1880:(whereas
1596:conjugate
1522:−
1498:−
1346:‖
1340:‖
1334:‖
1319:‖
1262:that map
1244:homothecy
1107:∣
1089:
1083:∈
1054:
1028:, of the
948:rotations
899:Lie group
891:transpose
854:composing
834:, is the
701:Conformal
589:Euclidean
196:nilpotent
12565:(1978),
12144:See also
11898:) → PSO(
11615:, where
11582:for the
10612:ordinary
10589:, where
10569:, where
10552:are the
10194:post hoc
9822:, where
8624: ;
8449:, where
8118:. Thus
7047:, where
6900:, where
6862:, where
6439:− 1) / 2
6326:+ ... +
5991:and the
5829:U/O ≅ Ω(
5254:SO(3) =
4762:Via the
3749:Topology
3666:) → SO(2
3528:, where
3296:) × {±1}
3021:, where
2974:− 1) / 2
2804:+ 1) / 2
2738:has two
2728:− 1) / 2
2522:) = cos(
2344:− 1) × (
2248:rotation
1841:is odd,
1772:{−1, +1}
914:has two
901:. It is
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
12702:, 2001
12667:1189139
12623:1859189
12087:basis.
12073:-frames
12045:is the
11880:) → SO(
11854:) → PO(
11680:) = SO(
11676:) ∩ GL(
11672:) = CO(
11646:) = O(2
11610:+ 1) ×
10938:is the
10912:) ⊂ O(2
10327:is the
10208:torsors
10118:to the
8473:{1, −1}
6276:is the
6274:O(3, 1)
6153:inertia
6123:, with
5663:be the
5444:is the
5362:may be
5296:(SO(4))
5215:, from
4025:Spin(2)
4004:is its
3890:-point
3881:O(1) =
3831:Please
3768:may be
3541:is the
3188:is the
3026:= SO(2)
2742:, with
2711:compact
2462:-sphere
2453:of the
2449:is the
2338:is the
2092:, ...,
1878:abelian
1801:(where
1700:equals
1246:), the
1180:over a
903:compact
887:inverse
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
12673:
12665:
12655:
12637:
12621:
12611:
12587:
12577:
12505:
12495:
12452:
12442:
12174:SO(3,
11948:, and
11835:) → O(
11819:) → O(
11797:and a
11724:, and
11546:angles
11540:Being
10908:) ⊂ U(
10892:) ⊂ U(
10649:(⋅, ⋅)
10147:square
10051:center
9994:where
9845:kernel
9772:|
9760:|
8964:where
8594:where
8086:, and
8054:with
7434:, and
7357:where
7222:, and
6983:φ
6846:where
6805:or to
6501:SO(2,
6479:, the
6477:> 2
6199:) = O(
6180:) = O(
6131:, and
6069:branes
6050:String
5898:String
5798:O) ≅ π
4331:) → O(
4008:. For
3971:> 2
3966:, for
3874:spaces
3543:kernel
3298:. The
3247:and a
3153:. In
2959:) = −1
2817:is an
2540:
2484:circle
2473:sphere
2334:where
1893:> 2
1768:kernel
1574:kernel
1394:. The
1184:, the
1006:. The
897:and a
885:whose
816:, the
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
12248:Notes
12185:SO(8)
12109:<
11934:Spin(
11918:PSO(2
11910:PSO(2
11876:Spin(
11760:(the
11742:(the
11590:. If
10630:Spin(
10335:from
10318:Here
10143:up to
10132:(the
9766:+ 1,
9114:with
8760:with
8453:is a
7403:O(2,
7236:. As
7063:O(2,
7058:Proof
7031:is a
7025:O(2,
6856:+ 1,
6731:) ≠ 0
6640:is a
6441:over
6052:from
6033:from
6014:from
5835:, so
5821:(U/O)
5806:O) =
5604:O) =
5580:O) =
5556:O) =
5536:O) =
5511:have
5506:Sp ×
5502:Sp =
5437:O) =
5421:over
5417:is a
5284:) = π
5235:) = π
5202:) = π
4788:) = π
4420:) = π
4386:From
4377:>
4303:acts
4201:is a
3998:Spin(
3931:SO(4)
3916:SO(3)
3902:SO(2)
3266:} by
3151:SO(2)
2982:) = 1
2951:) = 1
2925:is a
2492:SO(2)
2482:of a
2464:(for
2246:is a
2244:SO(3)
2167:with
1928:SO(2)
1874:SO(2)
1835:. If
1453:from
1375:of a
1250:from
1236:up to
1234:are,
1182:field
1176:on a
981:, an
965:SO(4)
959:SO(3)
953:SO(2)
922:is a
842:of a
836:group
730:Sp(∞)
727:SU(∞)
140:image
18:SO(n)
12653:ISBN
12635:ISBN
12609:ISBN
12575:ISBN
12493:ISBN
12440:ISBN
12319:SO(2
12315:+ 1)
12307:SO(2
12278:) ∩
12104:for
12033:The
12006:Spin
11950:PSO(
11930:+ 1)
11926:SO(2
11922:+ 1)
11825:and
11789:nor
11684:) ×
11668:CSO(
11660:CSO(
11650:) ×
11642:CO(2
11602:CO(2
11556:and
10932:)/O(
10904:USp(
10902:and
10873:curl
10428:and
10412:The
10108:The
10049:The
10041:) ·
10021:− 2·
9989:) ·
9928:and
9184:and
8811:and
8609:and
8430:+ 1)
8306:are
8302:and
8233:) ↦
7641:and
7373:= ±1
7368:and
7269:and
7240:and
7008:and
6744:The
6530:are
6418:and
6213:nor
6054:Spin
6031:Spin
5916:Spin
5718:P =
5708:P =
5617:Let
5591:spin
5497:and
5492:O ×
5488:O =
5476:and
5258:P =
5250:spin
5221:O(2)
4372:for
4344:− 1)
4335:+ 1)
4301:+ 1)
4211:+ 1)
4205:for
3973:the
3886:, a
3739:SO(2
3678:+ 1)
3674:SO(2
3670:+ 1)
3662:SO(2
3654:SO(2
3646:SO(2
3526:+ 1)
3522:SO(2
3476:<
3433:SO(2
3313:{±1}
3288:× {1
3283:and
3281:{±1}
3264:× {1
3257:{±1}
3253:{±1}
3186:+ 1)
3182:SO(2
3176:The
3167:+ 1)
3163:SO(2
3161:and
3159:+ 1)
3041:SO(2
3039:and
2978:det(
2955:det(
2947:det(
2701:and
2530:sin(
2526:) +
2515:exp(
2496:real
2488:O(2)
2478:The
2460:− 1)
2405:The
2398:The
2348:− 1)
1926:and
1924:O(2)
1861:and
1572:The
1428:) =
1413:∈ E(
1363:Let
1207:Name
962:and
724:O(∞)
713:Loop
532:and
12671:Zbl
12585:Zbl
12503:Zbl
12450:Zbl
12114:of
12075:).
12062:of
12010:PSO
11942:SO(
11894:SO(
11827:Pin
11811:Pin
11624:∖{0
11576:CO(
11102:SU(
10620:SO(
10604:SO(
10594:= 2
10578:+ 1
10574:= 2
10542:or
10496:or
10430:SO(
10358:of
10233:of
10188:of
10166:/ (
10161:in
10126:/ (
10072:+ 1
10058:= −
10016:to
9964:to
9875:SO(
9791:/ 2
9762:O(2
9758:of
9192:sin
9122:cos
9078:cos
9067:sin
9051:sin
9040:cos
8457:of
8441:+ 1
8371:= 1
8288:O(2
8258:O(2
7544:If
7386:of
7366:= 1
7284:= 1
7266:εωb
7248:in
7231:= –
7220:= 0
7218:ωbd
7209:= 1
7198:= Q
7196:AQA
7052:= ±
7010:O(2
6998:O(2
6920:to
6893:or
6885:to
6852:O(2
6741:).
6737:in
6718:is
6714:to
6648:to
6507:is
6495:is
6485:SO(
6483:of
6462:SO(
6458:≥ 2
6420:SO(
6393:SO(
6379:in
6314:in
6306:of
6259:SO(
6234:SO(
5843:1+7
5827:as
5584:/ 2
5560:/ 2
5339:).
5266:/ 2
5262:/ (
5243:/ 2
5210:/ 2
4814:= Ω
4793:+ 1
4414:+ 8
4381:+ 1
4366:(O(
4356:(O(
4337:is
4031:).
4013:= 2
3989:is
3979:SO(
3977:of
3937:by
3933:is
3919:is
3904:is
3888:two
3835:to
3292:O(2
3180:of
3155:O(2
3033:O(2
3031:In
2986:SO(
2984:is
2953:or
2744:SO(
2716:of
2703:SO(
2619:cos
2602:sin
2583:sin
2563:cos
2536:of
2507:(1)
2486:is
2469:= 3
2414:↦ −
2376:/ 2
2236:= 3
2202:SO(
2176:= 1
1909:of
1882:SO(
1876:is
1855:SO(
1853:of
1829:SO(
1813:of
1780:SO(
1778:of
1752:SO(
1736:of
1726:SO(
1712:or
1696:of
1613:of
1586:of
1576:of
1569:).
1461:to
1287:GL(
1256:to
1172:or
1033:GL(
932:SO(
838:of
812:In
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
12731::
12698:,
12692:,
12669:,
12663:MR
12661:,
12619:MR
12617:,
12607:,
12599:,
12583:,
12573:,
12501:,
12475:^
12448:.
12426:^
12410:.
12406:.
12294:±1
12286:,
12280:GL
12274:O(
12039:O(
12014:SO
12004:–
11960:,
11940:,
11892:,
11874:,
11850:O(
11848:,
11809:,
11732:.
11720:,
11713:.
11689:.
11655:.
11638:±1
11620:=
11600::
11507:16
11470:12
11433:10
11224:O(
11100:;
11094:or
11086:U(
11084:–
10942:.
10928:U(
10888:O(
10868:.
10862:,
10599:.
10448:×
10434:,
10422:,
10418:O(
10383:(μ
10340:(O
10237:.
10214:.
10172:.
10033:)/
10029:,
9981:)/
9977:,
9969:+
9921:.
9915:±1
9907:,
9903:O(
9891:,
9887:O(
9879:,
9853:,
9849:O(
9838:−1
9816:−
8995:SO
8652:SO
8468:,
8292:,
8283:.
8262:,
8253:.
8239:αb
8237:+
8229:,
8104:+
8091:=
8074:ωb
8072:+
8059:=
7445:αb
7443:–
7439:=
7431:αb
7429:+
7425:=
7379:.
7364:ωb
7362:–
7276:εa
7274:=
7264:=
7229:ωd
7227:–
7216:–
7214:ac
7211:,
7207:ωb
7205:–
7054:.
7041:−
7037:2(
7020:.
7014:,
7002:,
6910:−1
6875:−1
6848:𝜑
6513:.
6489:,
6466:,
6424:,
6412:,
6408:O(
6403:.
6397:,
6385:,
6381:O(
6369:,
6365:O(
6350:,
6346:O(
6269:.
6263:,
6250:,
6246:O(
6238:,
6226:,
6222:O(
6195:O(
6190:.
6184:,
6176:,
6172:O(
6164:,
6160:O(
6145:,
6137:−1
6114:=
6110:+
6071:.
6037:,
6035:SO
6018:,
6012:SO
5934:SO
5863::
5851:.
5849:O)
5833:O)
5788:,
5778:O)
5764:O)
5750:O)
5736:O)
5713:,
5703:,
5698:=
5690:,
5685:=
5646:,
5638:,
5632:,
5481:/U
5470:,
5448:.
5415:O)
5311:,
4818:×
4810:×
4808:BO
4806:=
4804:KO
4797:BO
4396:≅
4370:))
4327:O(
4297:O(
4217::
4207:O(
3983:,
3948:×
3940:SU
3876::
3745:.
3735:−1
3640:−1
3537:−1
3428:.
3422:±1
3364:or
3000:A
2992:.
2939:O(
2919:O(
2811:O(
2786:=
2768:O(
2758:.
2732:O(
2695:O(
2685:.
2675:O(
2659:O(
2443:O(
2421:O(
2389:O(
2378:.
2358:.
2263:O(
2228:.
2222:+1
2214:−1
2210:−1
2197:.
2178:.
2172:+
1940:O(
1930:.
1869:.
1863:{±
1843:O(
1815:O(
1795:{±
1790:.
1760:O(
1738:O(
1714:−1
1629:.
1623:O(
1615:O(
1606:.
1600:O(
1588:E(
1463:O(
1455:E(
1435:O(
1403:∈
1365:E(
1291:,
1276:O(
1238:a
1228:O(
1086:GL
1037:,
1022:,
1018:O(
1011:×
986:×
970:−1
956:,
905:.
869:×
828:O(
615:U(
591:E(
579:O(
37:→
12531:(
12518:(
12466:(
12456:.
12420:.
12323:)
12321:k
12313:k
12309:k
12290:)
12288:Z
12284:n
12282:(
12276:n
12178:)
12176:R
12137:k
12131:k
12121:k
12111:n
12107:k
12102:)
12100:R
12098:(
12095:k
12093:V
12070:n
12060:)
12058:R
12056:(
12053:n
12051:V
12043:)
12041:n
11992:)
11988:R
11984:,
11981:n
11978:(
11973:o
11970:s
11954:)
11952:n
11946:)
11944:n
11938:)
11936:n
11928:k
11920:k
11914:)
11912:k
11904:.
11902:)
11900:n
11896:n
11886:,
11884:)
11882:n
11878:n
11860:.
11858:)
11856:n
11852:n
11841:,
11839:)
11837:n
11833:n
11831:(
11829:−
11823:)
11821:n
11817:n
11815:(
11813:+
11776:.
11768:n
11765:B
11754:)
11750:n
11747:A
11686:R
11682:n
11678:n
11674:n
11670:n
11664:)
11662:n
11652:R
11648:k
11644:k
11633:n
11622:R
11618:R
11612:R
11608:k
11604:k
11593:n
11580:)
11578:n
11571:R
11510:)
11504:(
11500:O
11487:8
11482:E
11473:)
11467:(
11463:O
11450:7
11445:E
11436:)
11430:(
11426:O
11413:6
11408:E
11399:)
11396:9
11393:(
11389:O
11376:4
11371:F
11362:)
11359:3
11356:(
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1076:{
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571:)
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499:)
486:Z
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461:Z
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415:(
328:p
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285:n
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272:n
269:A
261:n
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247:Z
20:)
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