741:
49:
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and
570:
3726:
1700:
2358:
1977:
3959:
399:
3801:
1468:, meaning that all its maps are surjective. Without loss of generality, it suffices to consider only surjective systems since given any inverse system, it is possible to first construct its profinite group
3660:
530:
1606:
3455:
1283:
2139:
2226:
1877:
1007:
2441:
1817:
2865:
2401:
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1781:
298:
76:
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1063:
846:
1921:
565:
1842:
3164:
246:
1315:
1139:
431:
3616:
2010:
1729:
3385:
3114:
3353:
3184:
3134:
3871:
1162:
1083:
462:
908:
2745:
2274:
2089:
2853:
2822:
2799:
2714:
2580:
2553:
2486:
1489:
877:
814:
3844:
3821:
3748:
3585:
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3517:
3497:
3477:
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3327:
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3229:
3091:
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2988:
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1749:
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1406:
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1103:
1027:
952:
928:
147:
96:
4138:
Nikolov, Nikolay; Segal, Dan (2007). "On finitely generated profinite groups. I: Strong completeness and uniform bounds. II: Products in quasisimple groups".
736:{\displaystyle \varprojlim G_{i}=\left\{(g_{i})_{i\in I}\in {\textstyle \prod \limits _{i\in I}}G_{i}:f_{i}^{j}(g_{j})=g_{i}{\text{ for all }}i\leq j\right\}}
3665:
1635:
4110:
2283:
4506:
1347:
satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit
4392:
1926:
1240:
is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique
4405:
2658:. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is
3240:
3905:
310:
3753:
3186:
is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its
2908:
3621:
4444:
4295:
4249:
99:
4313:
Nikolov, Nikolay; Segal, Dan (2007), "On finitely generated profinite groups, I: strong completeness and uniform bounds",
467:
4484:
4060:
3548:
1569:
4346:
Nikolov, Nikolay; Segal, Dan (2007), "On finitely generated profinite groups, II: products in quasisimple groups",
3419:
1247:
3997:
2654:
of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the
2098:
4236:. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin, Heidelberg: Springer Berlin Heidelberg.
2188:
1847:
968:
4554:
3093:
are topologically finitely generated profinite groups that are isomorphic as discrete groups by an isomorphism
2669:
subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the
2406:
1791:
1241:
17:
4265:
3285:
profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian
2363:
4121:
98:
elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are
4559:
2876:
2637:
2015:
1754:
962:
by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).
255:
175:
167:
53:
3876:
2617:
are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an
2614:
1536:
1191:
3356:
3268:
2921:
1350:
1036:
819:
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535:
4383:
4286:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.).
4085:
2651:
1822:
1452:
is topologically finitely generated then it is in addition equal to its own profinite completion.
4006:
3414:
3139:
2872:, in any topologically finitely generated profinite group (that is, a profinite group that has a
1106:
171:
216:
3208:
2556:
1288:
1112:
404:
121:
of the resulting profinite group; in a sense, these quotients approximate the profinite group.
3598:
2063:
of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if
1982:
1705:
4501:
3544:
3361:
3236:
3096:
2607:
2045:
1324:
Any group constructed by the first definition satisfies the axioms in the second definition.
3332:
3169:
3119:
2629:, however, are in general not profinite: for any prescribed group, there is a 2-dimensional
4454:
3974:
3969:
3849:
3264:
1147:
1068:
778:
440:
4537:
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4462:
4305:
4177:
2879:) the subgroups of finite index are open. This generalizes an earlier analogous result of
882:
8:
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955:
114:
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1471:
859:
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on the category of profinite groups. Further, being profinite is an extension property.
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3721:{\displaystyle \langle \sigma \rangle =\left\{\sigma ^{n}:n\in \mathbb {Z} \right\}.}
3556:
2880:
2748:
2622:
2618:
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1786:
1505:
762:
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301:
39:
27:
Topological group that is in a certain sense assembled from a system of finite groups
4169:
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4515:
4490:
4458:
4414:
4365:
4332:
4301:
4237:
4173:
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2857:
A subgroup of a profinite group is open if and only if it is closed and has finite
2655:
2606:
in one variable over the complex numbers.) Not every profinite group occurs as an
2092:
747:
113:
between them. Without loss of generality, these homomorphisms can be assumed to be
4419:
1731:
The topology on this profinite group is the same as the topology arising from the
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4450:
4436:
4379:
4287:
2060:
1409:
931:
758:
156:
4370:
4337:
4161:
1695:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} \to \mathbb {Z} /p^{m}\mathbb {Z} }
2450:
2445:
1629:
249:
207:
203:
118:
103:
4241:
2353:{\displaystyle \operatorname {Gal} (F_{1}/K)\to \operatorname {Gal} (F_{2}/K)}
4548:
3286:
3282:
2825:
2659:
2056:
1512:
959:
774:
199:
163:
129:
125:
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2717:
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where the intersection runs through all normal subgroups of finite index).
434:
211:
152:
43:
2747:
is profinite; the topology arising from the profiniteness agrees with the
3979:
2884:
782:
195:
31:
4360:
4327:
1972:{\displaystyle \mathbb {Z} /n\mathbb {Z} \to \mathbb {Z} /m\mathbb {Z} }
4529:
2666:
2630:
2461:
profinite group is isomorphic to one arising from the Galois theory of
2280:
Galois extension. For the limit process, the restriction homomorphisms
109:
To construct a profinite group one needs a system of finite groups and
4435:, Lecture Notes in Mathematics (in French), vol. 5 (5 ed.),
4152:
4036:
2873:
2869:
1030:
4030:
Segal, Dan (2007-03-29). "Some aspects of profinite group theory".
3271:
2829:
3010:
is topologically finitely generated. Indeed, any open subgroup of
1460:
In practice, the inverse system of finite groups is almost always
186:
Profinite groups can be defined in either of two equivalent ways.
3954:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} ,n\in \mathbb {N} .}
394:{\displaystyle \{f_{i}^{j}:G_{j}\to G_{i}\mid i,j\in I,i\leq j\}}
3796:{\displaystyle G\cong {\textstyle \prod \limits _{p\in S}}G_{p}}
3329:
is projective if for every surjective morphism from a profinite
210:
finite groups. In this context, an inverse system consists of a
3274:
is finite. This is equivalent, in fact, to being 'ind-finite'.
781:
topological group: that is, a topological group that is also a
3136:
is bijective and continuous by the above result. Furthermore,
2602:
the inverse Galois problem is settled, such as the field of
2185:
fixed. This group is the inverse limit of the finite groups
2615:étale fundamental groups considered in algebraic geometry
170:. A non-compact generalization of the concept is that of
3655:{\displaystyle G={\overline {\langle \sigma \rangle }},}
3309:
for every extension. This is equivalent to saying that
2914:
As an easy corollary of the
Nikolov–Segal result above,
1844:
In detail, it is the inverse limit of the finite groups
78:
such that every group in the system can be generated by
3543:
Every projective profinite group can be realized as an
3764:
638:
464:
and the collection satisfies the composition property
42:
that is in a certain sense assembled from a system of
3908:
3879:
3852:
3832:
3809:
3756:
3736:
3668:
3624:
3601:
3595:
if it is topologically generated by a single element
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84:
56:
4011:
Pages displaying wikidata descriptions as a fallback
4002:
Pages displaying wikidata descriptions as a fallback
3988:
Pages displaying wikidata descriptions as a fallback
2801:
which allows us to measure the "size" of subsets of
753:
One can also define the inverse limit in terms of a
3050:
is also of finite index, and hence it must be open.
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525:{\displaystyle f_{i}^{j}\circ f_{j}^{k}=f_{i}^{k}}
524:
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393:
292:
240:
189:
141:
90:
70:
4424:. Review of several books about profinite groups.
2693:is a closed normal subgroup of a profinite group
879:It is defined as the inverse limit of the groups
194:A profinite group is a topological group that is
117:, in which case the finite groups will appear as
4546:
4507:Proceedings of the American Mathematical Society
2508:will be in this case. In fact, for many fields
1566:). It is the inverse limit of the finite groups
768:
3409:is equivalent to either of the two properties:
3243:.) The usual terminology is different: a group
124:Important examples of profinite groups are the
4504:(1974), "Profinite groups are Galois groups",
3292:
2248:ranges over all intermediate fields such that
1601:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
4514:(2), American Mathematical Society: 639–640,
4406:Bulletin of the American Mathematical Society
3450:{\displaystyle \operatorname {cd} (G)\leq 1;}
2528:one does not know in general precisely which
1278:{\displaystyle g:{\widehat {G}}\rightarrow H}
4345:
4312:
4137:
3675:
3669:
3640:
3634:
2012:This group is the product of all the groups
388:
314:
284:
259:
4281:
3239:of finite groups. (In particular, it is an
2134:{\displaystyle G=\operatorname {Gal} (L/K)}
4500:
4220:Fried & Jarden (2008) pp. 208,545
2454:
2221:{\displaystyle \operatorname {Gal} (F/K),}
1872:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
1504:Finite groups are profinite, if given the
1002:{\displaystyle \eta :G\to {\widehat {G}},}
4519:
4418:
4369:
4359:
4336:
4326:
4282:Fried, Michael D.; Jarden, Moshe (2008).
4151:
4035:
3944:
3930:
3910:
3882:
3706:
2488:but one cannot (yet) control which field
2436:{\displaystyle \operatorname {Gal} (L/K)}
2021:
1995:
1989:
1965:
1952:
1944:
1931:
1865:
1852:
1827:
1812:{\displaystyle {\widehat {\mathbb {Z} }}}
1799:
1760:
1688:
1668:
1660:
1640:
1594:
1574:
1542:
64:
4399:
4229:
4378:
4266:"MO. decomposition of procyclic groups"
4198:
4196:
3030:is of finite index, so its preimage in
2918:surjective discrete group homomorphism
2644:
788:
14:
4547:
4050:
2909:classification of finite simple groups
1188:and any continuous group homomorphism
1085:is injective if and only if the group
4467:
4427:
4211:Fried & Jarden (2008) p. 207
4190:Fried & Jarden (2008) p. 497
4111:"Inverse limits and profinite groups"
4029:
2883:for topologically finitely generated
2774:is compact Hausdorff, there exists a
2396:{\displaystyle F_{2}\subseteq F_{1}.}
1562:under addition is profinite (in fact
1455:
4205:
4193:
4184:
4108:
4104:
4102:
3194:
1563:
1495:it as its own profinite completion.
4083:
3766:
3389:Projectivity for a profinite group
1432:ordered by (reverse) inclusion. If
816:there is a related profinite group
761:terms, this is a special case of a
640:
24:
4009: – type of mathematical group
3562:
3211:to profinite groups; i.e. a group
2633:whose fundamental group equals it.
1164:is characterized by the following
25:
4571:
4521:10.1090/S0002-9939-1974-0325587-3
4099:
3549:pseudo algebraically closed field
2037:{\displaystyle \mathbb {Z} _{p},}
1776:{\displaystyle \mathbb {Z} _{p}.}
293:{\displaystyle \{G_{i}:i\in I\},}
71:{\displaystyle d\in \mathbb {N} }
3895:{\displaystyle \mathbb {Z} _{p}}
1555:{\displaystyle \mathbb {Z} _{p}}
1213:{\displaystyle f:G\rightarrow H}
965:There is a natural homomorphism
4258:
4223:
4118:University of California, Davis
4000: – concept in group theory
3998:Residual property (mathematics)
3986: – type of profinite group
2943:{\displaystyle \varphi :G\to H}
1819:is the profinite completion of
1381:{\displaystyle \varprojlim G/N}
1058:{\displaystyle {\widehat {G}}.}
841:{\displaystyle {\widehat {G}},}
190:First definition (constructive)
4214:
4131:
4077:
4044:
4023:
3435:
3429:
3368:
3339:
2934:
2430:
2416:
2347:
2326:
2317:
2314:
2293:
2212:
2198:
2128:
2114:
1991:
1948:
1916:{\displaystyle n=1,2,3,\dots }
1664:
1319:
1269:
1204:
981:
698:
685:
619:
605:
567:The inverse limit is the set:
560:{\displaystyle i\leq j\leq k.}
345:
232:
220:
13:
1:
4475:, Translated by Patrick Ion,
4420:10.1090/S0273-0979-01-00914-4
4016:
1837:{\displaystyle \mathbb {Z} .}
769:Second definition (axiomatic)
181:
4051:Wilson, John Stuart (1998).
3750:is procyclic if and only if
3662:the closure of the subgroup
3644:
2754:Since every profinite group
2636:The automorphism group of a
2532:occur as Galois groups over
1168:: given any profinite group
958:(these normal subgroups are
174:. Even more general are the
7:
4371:10.4007/annals.2007.165.239
4338:10.4007/annals.2007.165.171
4162:10.4007/annals.2007.165.171
4055:. Oxford: Clarendon Press.
3963:
3293:Projective profinite groups
3231:is ind-finite if it is the
3159:{\displaystyle \iota ^{-1}}
2877:finitely generated subgroup
1498:
1029:under this homomorphism is
176:totally disconnected groups
10:
4576:
3207:, which is the conceptual
2864:According to a theorem of
2638:locally finite rooted tree
2165:that keep all elements of
241:{\displaystyle (I,\leq ),}
4242:10.1007/978-3-662-03983-0
4230:Neukirch, Jürgen (1999).
2990:is continuous as long as
2950:between profinite groups
2403:The topology obtained on
1310:{\displaystyle f=g\eta .}
1134:{\displaystyle \cap N=1,}
793:Given an arbitrary group
426:{\displaystyle f_{i}^{i}}
162:Every profinite group is
3873:is isomorphic to either
3823:ranges over some set of
3611:{\displaystyle \sigma ;}
3551:. This result is due to
2005:{\displaystyle m\,|\,n.}
1724:{\displaystyle n\geq m.}
1408:ranges through the open
172:locally profinite groups
4433:Cohomologie galoisienne
4403:(2001), "Book Review",
4233:Algebraic Number Theory
4202:Serre (1997) p. 58
4146:(1): 171–238, 239–273.
4007:Residually finite group
3415:cohomological dimension
3380:{\displaystyle G\to H.}
3166:is also continuous, so
3109:{\displaystyle \iota .}
773:A profinite group is a
4502:Waterhouse, William C.
3955:
3896:
3867:
3840:
3817:
3797:
3744:
3722:
3656:
3612:
3581:
3533:
3513:
3493:
3473:
3451:
3403:
3381:
3349:
3348:{\displaystyle H\to G}
3323:
3257:
3225:
3180:
3179:{\displaystyle \iota }
3160:
3130:
3129:{\displaystyle \iota }
3110:
3087:
3067:
3044:
3024:
3004:
2984:
2964:
2944:
2899:
2849:
2818:
2795:
2768:
2741:
2710:
2687:
2596:
2576:
2557:inverse Galois problem
2549:
2522:
2502:
2482:
2437:
2397:
2354:
2270:
2242:
2222:
2179:
2159:
2135:
2085:
2038:
2006:
1973:
1917:
1873:
1838:
1813:
1777:
1745:
1725:
1696:
1622:
1602:
1556:
1526:
1485:
1446:
1426:
1402:
1382:
1341:
1327:Conversely, any group
1311:
1279:
1234:
1214:
1182:
1158:
1135:
1099:
1079:
1059:
1023:
1003:
948:
924:
904:
873:
842:
810:
737:
561:
526:
458:
427:
395:
294:
242:
143:
92:
72:
4555:Infinite group theory
4348:Annals of Mathematics
4315:Annals of Mathematics
3956:
3897:
3868:
3866:{\displaystyle G_{p}}
3841:
3818:
3798:
3745:
3723:
3657:
3613:
3582:
3545:absolute Galois group
3534:
3514:
3494:
3474:
3452:
3404:
3382:
3350:
3324:
3297:A profinite group is
3258:
3226:
3199:There is a notion of
3181:
3161:
3131:
3111:
3088:
3068:
3045:
3025:
3005:
2985:
2965:
2945:
2907:. The proof uses the
2900:
2850:
2819:
2796:
2769:
2742:
2711:
2688:
2608:absolute Galois group
2597:
2577:
2550:
2523:
2503:
2483:
2438:
2398:
2355:
2271:
2243:
2223:
2180:
2160:
2136:
2095:, consider the group
2086:
2046:absolute Galois group
2039:
2007:
1974:
1923:with the modulo maps
1918:
1874:
1839:
1814:
1778:
1746:
1726:
1697:
1632:and the natural maps
1623:
1603:
1557:
1527:
1486:
1447:
1427:
1403:
1383:
1342:
1312:
1280:
1235:
1215:
1183:
1159:
1157:{\displaystyle \eta }
1136:
1100:
1080:
1078:{\displaystyle \eta }
1060:
1024:
1004:
949:
925:
905:
874:
843:
811:
738:
562:
527:
459:
457:{\displaystyle G_{i}}
428:
396:
295:
243:
144:
93:
73:
50:only if there exists
3975:Locally cyclic group
3970:Hausdorff completion
3906:
3877:
3850:
3830:
3807:
3754:
3734:
3730:A topological group
3666:
3622:
3599:
3571:
3523:
3503:
3483:
3463:
3420:
3393:
3362:
3333:
3313:
3247:
3215:
3170:
3140:
3120:
3097:
3077:
3057:
3034:
3014:
2994:
2974:
2954:
2922:
2889:
2836:
2805:
2782:
2758:
2723:
2697:
2677:
2645:Properties and facts
2586:
2563:
2536:
2512:
2492:
2469:
2407:
2364:
2284:
2252:
2232:
2189:
2169:
2149:
2099:
2067:
2016:
1983:
1927:
1883:
1848:
1823:
1792:
1755:
1735:
1706:
1636:
1612:
1570:
1537:
1516:
1472:
1436:
1416:
1392:
1351:
1331:
1289:
1248:
1224:
1192:
1172:
1148:
1113:
1089:
1069:
1037:
1013:
969:
938:
914:
903:{\displaystyle G/N,}
883:
860:
852:profinite completion
820:
797:
789:Profinite completion
779:totally disconnected
571:
536:
468:
441:
405:
311:
256:
217:
168:totally disconnected
133:
82:
54:
4401:Lubotzky, Alexander
3281:, one can see that
2740:{\displaystyle G/N}
2269:{\displaystyle F/K}
2143:field automorphisms
2084:{\displaystyle L/K}
1751:-adic valuation on
1244:group homomorphism
716: for all
684:
521:
503:
485:
422:
331:
155:of infinite-degree
111:group homomorphisms
4560:Topological groups
4469:Serre, Jean-Pierre
4429:Serre, Jean-Pierre
4086:"Profinite Groups"
4084:Lenstra, Hendrik.
3951:
3892:
3863:
3836:
3813:
3793:
3781:
3780:
3740:
3718:
3652:
3608:
3577:
3567:A profinite group
3553:Alexander Lubotzky
3529:
3509:
3489:
3469:
3447:
3399:
3377:
3345:
3319:
3279:Pontryagin duality
3269:finitely generated
3253:
3221:
3176:
3156:
3126:
3106:
3083:
3063:
3040:
3020:
3000:
2980:
2960:
2940:
2895:
2848:{\displaystyle G.}
2845:
2817:{\displaystyle G,}
2814:
2794:{\displaystyle G,}
2791:
2764:
2737:
2709:{\displaystyle G,}
2706:
2683:
2627:algebraic topology
2623:fundamental groups
2604:rational functions
2592:
2575:{\displaystyle K.}
2572:
2548:{\displaystyle K.}
2545:
2518:
2498:
2481:{\displaystyle K,}
2478:
2433:
2393:
2350:
2266:
2238:
2218:
2175:
2155:
2141:consisting of all
2131:
2081:
2034:
2002:
1969:
1913:
1869:
1834:
1809:
1787:profinite integers
1773:
1741:
1721:
1692:
1618:
1598:
1552:
1522:
1484:{\displaystyle G,}
1481:
1456:Surjective systems
1442:
1422:
1398:
1378:
1362:
1337:
1307:
1275:
1230:
1210:
1178:
1166:universal property
1154:
1131:
1095:
1075:
1055:
1019:
999:
944:
920:
900:
872:{\displaystyle G.}
869:
838:
809:{\displaystyle G,}
806:
755:universal property
743:equipped with the
733:
670:
655:
654:
582:
557:
522:
507:
489:
471:
454:
423:
408:
391:
317:
304:, and a family of
290:
238:
139:
100:Lagrange's theorem
88:
68:
4473:Galois cohomology
4446:978-3-540-58002-7
4297:978-3-540-77269-9
4251:978-3-642-08473-7
4109:Osserman, Brian.
4093:Leiden University
3993:Profinite integer
3839:{\displaystyle S}
3816:{\displaystyle p}
3765:
3743:{\displaystyle G}
3647:
3580:{\displaystyle G}
3557:Lou van den Dries
3532:{\displaystyle p}
3512:{\displaystyle G}
3492:{\displaystyle p}
3472:{\displaystyle p}
3402:{\displaystyle G}
3322:{\displaystyle G}
3256:{\displaystyle G}
3224:{\displaystyle G}
3195:Ind-finite groups
3086:{\displaystyle H}
3066:{\displaystyle G}
3043:{\displaystyle G}
3023:{\displaystyle H}
3003:{\displaystyle G}
2983:{\displaystyle H}
2963:{\displaystyle G}
2898:{\displaystyle p}
2881:Jean-Pierre Serre
2767:{\displaystyle G}
2749:quotient topology
2686:{\displaystyle N}
2671:subspace topology
2619:algebraic variety
2595:{\displaystyle K}
2582:(For some fields
2559:for a field
2521:{\displaystyle K}
2501:{\displaystyle K}
2455:Waterhouse (1974)
2241:{\displaystyle F}
2178:{\displaystyle K}
2158:{\displaystyle L}
1806:
1744:{\displaystyle p}
1621:{\displaystyle n}
1525:{\displaystyle p}
1506:discrete topology
1445:{\displaystyle G}
1425:{\displaystyle G}
1401:{\displaystyle N}
1355:
1340:{\displaystyle G}
1266:
1233:{\displaystyle G}
1181:{\displaystyle H}
1144:The homomorphism
1107:residually finite
1098:{\displaystyle G}
1065:The homomorphism
1049:
1022:{\displaystyle G}
1009:and the image of
993:
960:partially ordered
947:{\displaystyle G}
930:runs through the
923:{\displaystyle N}
832:
717:
639:
575:
302:discrete topology
252:of finite groups
142:{\displaystyle p}
91:{\displaystyle d}
40:topological group
16:(Redirected from
4567:
4540:
4523:
4497:
4465:
4423:
4422:
4395:
4391:, talk given at
4390:
4385:Profinite Groups
4380:Lenstra, Hendrik
4374:
4373:
4363:
4341:
4340:
4330:
4309:
4284:Field arithmetic
4274:
4273:
4262:
4256:
4255:
4227:
4221:
4218:
4212:
4209:
4203:
4200:
4191:
4188:
4182:
4181:
4155:
4135:
4129:
4128:
4126:
4120:. Archived from
4115:
4106:
4097:
4096:
4090:
4081:
4075:
4074:
4053:Profinite groups
4048:
4042:
4041:
4039:
4027:
4012:
4003:
3989:
3960:
3958:
3957:
3952:
3947:
3933:
3928:
3927:
3918:
3913:
3901:
3899:
3898:
3893:
3891:
3890:
3885:
3872:
3870:
3869:
3864:
3862:
3861:
3845:
3843:
3842:
3837:
3822:
3820:
3819:
3814:
3802:
3800:
3799:
3794:
3792:
3791:
3782:
3779:
3749:
3747:
3746:
3741:
3727:
3725:
3724:
3719:
3714:
3710:
3709:
3695:
3694:
3661:
3659:
3658:
3653:
3648:
3643:
3632:
3617:
3615:
3614:
3609:
3593:
3592:
3586:
3584:
3583:
3578:
3538:
3536:
3535:
3530:
3518:
3516:
3515:
3510:
3498:
3496:
3495:
3490:
3478:
3476:
3475:
3470:
3459:for every prime
3456:
3454:
3453:
3448:
3408:
3406:
3405:
3400:
3386:
3384:
3383:
3378:
3354:
3352:
3351:
3346:
3328:
3326:
3325:
3320:
3307:lifting property
3303:
3302:
3262:
3260:
3259:
3254:
3237:inductive system
3230:
3228:
3227:
3222:
3205:
3204:
3203:ind-finite group
3185:
3183:
3182:
3177:
3165:
3163:
3162:
3157:
3155:
3154:
3135:
3133:
3132:
3127:
3115:
3113:
3112:
3107:
3092:
3090:
3089:
3084:
3072:
3070:
3069:
3064:
3049:
3047:
3046:
3041:
3029:
3027:
3026:
3021:
3009:
3007:
3006:
3001:
2989:
2987:
2986:
2981:
2969:
2967:
2966:
2961:
2949:
2947:
2946:
2941:
2904:
2902:
2901:
2896:
2854:
2852:
2851:
2846:
2824:compute certain
2823:
2821:
2820:
2815:
2800:
2798:
2797:
2792:
2773:
2771:
2770:
2765:
2746:
2744:
2743:
2738:
2733:
2715:
2713:
2712:
2707:
2692:
2690:
2689:
2684:
2656:product topology
2601:
2599:
2598:
2593:
2581:
2579:
2578:
2573:
2554:
2552:
2551:
2546:
2527:
2525:
2524:
2519:
2507:
2505:
2504:
2499:
2487:
2485:
2484:
2479:
2443:is known as the
2442:
2440:
2439:
2434:
2426:
2402:
2400:
2399:
2394:
2389:
2388:
2376:
2375:
2360:are used, where
2359:
2357:
2356:
2351:
2343:
2338:
2337:
2310:
2305:
2304:
2275:
2273:
2272:
2267:
2262:
2247:
2245:
2244:
2239:
2227:
2225:
2224:
2219:
2208:
2184:
2182:
2181:
2176:
2164:
2162:
2161:
2156:
2140:
2138:
2137:
2132:
2124:
2093:Galois extension
2090:
2088:
2087:
2082:
2077:
2061:field extensions
2043:
2041:
2040:
2035:
2030:
2029:
2024:
2011:
2009:
2008:
2003:
1994:
1978:
1976:
1975:
1970:
1968:
1960:
1955:
1947:
1939:
1934:
1922:
1920:
1919:
1914:
1878:
1876:
1875:
1870:
1868:
1860:
1855:
1843:
1841:
1840:
1835:
1830:
1818:
1816:
1815:
1810:
1808:
1807:
1802:
1797:
1782:
1780:
1779:
1774:
1769:
1768:
1763:
1750:
1748:
1747:
1742:
1730:
1728:
1727:
1722:
1701:
1699:
1698:
1693:
1691:
1686:
1685:
1676:
1671:
1663:
1658:
1657:
1648:
1643:
1628:ranges over all
1627:
1625:
1624:
1619:
1607:
1605:
1604:
1599:
1597:
1592:
1591:
1582:
1577:
1561:
1559:
1558:
1553:
1551:
1550:
1545:
1531:
1529:
1528:
1523:
1490:
1488:
1487:
1482:
1466:
1465:
1451:
1449:
1448:
1443:
1431:
1429:
1428:
1423:
1410:normal subgroups
1407:
1405:
1404:
1399:
1387:
1385:
1384:
1379:
1374:
1363:
1346:
1344:
1343:
1338:
1316:
1314:
1313:
1308:
1284:
1282:
1281:
1276:
1268:
1267:
1259:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1187:
1185:
1184:
1179:
1163:
1161:
1160:
1155:
1140:
1138:
1137:
1132:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1051:
1050:
1042:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
995:
994:
986:
953:
951:
950:
945:
932:normal subgroups
929:
927:
926:
921:
909:
907:
906:
901:
893:
878:
876:
875:
870:
854:
853:
847:
845:
844:
839:
834:
833:
825:
815:
813:
812:
807:
763:cofiltered limit
748:product topology
742:
740:
739:
734:
732:
728:
718:
715:
713:
712:
697:
696:
683:
678:
666:
665:
656:
653:
633:
632:
617:
616:
596:
595:
583:
566:
564:
563:
558:
531:
529:
528:
523:
520:
515:
502:
497:
484:
479:
463:
461:
460:
455:
453:
452:
432:
430:
429:
424:
421:
416:
400:
398:
397:
392:
357:
356:
344:
343:
330:
325:
300:each having the
299:
297:
296:
291:
271:
270:
247:
245:
244:
239:
157:field extensions
148:
146:
145:
140:
97:
95:
94:
89:
77:
75:
74:
69:
67:
21:
4575:
4574:
4570:
4569:
4568:
4566:
4565:
4564:
4545:
4544:
4487:
4477:Springer-Verlag
4447:
4437:Springer-Verlag
4388:
4361:math.GR/0604400
4328:math.GR/0604399
4298:
4288:Springer-Verlag
4278:
4277:
4264:
4263:
4259:
4252:
4228:
4224:
4219:
4215:
4210:
4206:
4201:
4194:
4189:
4185:
4136:
4132:
4124:
4113:
4107:
4100:
4088:
4082:
4078:
4063:
4049:
4045:
4028:
4024:
4019:
4010:
4001:
3987:
3966:
3943:
3929:
3923:
3919:
3914:
3909:
3907:
3904:
3903:
3886:
3881:
3880:
3878:
3875:
3874:
3857:
3853:
3851:
3848:
3847:
3831:
3828:
3827:
3808:
3805:
3804:
3787:
3783:
3769:
3763:
3755:
3752:
3751:
3735:
3732:
3731:
3705:
3690:
3686:
3685:
3681:
3667:
3664:
3663:
3633:
3631:
3623:
3620:
3619:
3600:
3597:
3596:
3590:
3589:
3572:
3569:
3568:
3565:
3563:Procyclic group
3524:
3521:
3520:
3504:
3501:
3500:
3484:
3481:
3480:
3464:
3461:
3460:
3421:
3418:
3417:
3394:
3391:
3390:
3363:
3360:
3359:
3334:
3331:
3330:
3314:
3311:
3310:
3300:
3299:
3295:
3248:
3245:
3244:
3216:
3213:
3212:
3202:
3201:
3197:
3171:
3168:
3167:
3147:
3143:
3141:
3138:
3137:
3121:
3118:
3117:
3098:
3095:
3094:
3078:
3075:
3074:
3058:
3055:
3054:
3035:
3032:
3031:
3015:
3012:
3011:
2995:
2992:
2991:
2975:
2972:
2971:
2955:
2952:
2951:
2923:
2920:
2919:
2890:
2887:
2886:
2866:Nikolay Nikolov
2837:
2834:
2833:
2806:
2803:
2802:
2783:
2780:
2779:
2759:
2756:
2755:
2729:
2724:
2721:
2720:
2698:
2695:
2694:
2678:
2675:
2674:
2647:
2587:
2584:
2583:
2564:
2561:
2560:
2537:
2534:
2533:
2513:
2510:
2509:
2493:
2490:
2489:
2470:
2467:
2466:
2422:
2408:
2405:
2404:
2384:
2380:
2371:
2367:
2365:
2362:
2361:
2339:
2333:
2329:
2306:
2300:
2296:
2285:
2282:
2281:
2258:
2253:
2250:
2249:
2233:
2230:
2229:
2204:
2190:
2187:
2186:
2170:
2167:
2166:
2150:
2147:
2146:
2120:
2100:
2097:
2096:
2073:
2068:
2065:
2064:
2025:
2020:
2019:
2017:
2014:
2013:
1990:
1984:
1981:
1980:
1964:
1956:
1951:
1943:
1935:
1930:
1928:
1925:
1924:
1884:
1881:
1880:
1864:
1856:
1851:
1849:
1846:
1845:
1826:
1824:
1821:
1820:
1798:
1796:
1795:
1793:
1790:
1789:
1764:
1759:
1758:
1756:
1753:
1752:
1736:
1733:
1732:
1707:
1704:
1703:
1687:
1681:
1677:
1672:
1667:
1659:
1653:
1649:
1644:
1639:
1637:
1634:
1633:
1630:natural numbers
1613:
1610:
1609:
1593:
1587:
1583:
1578:
1573:
1571:
1568:
1567:
1546:
1541:
1540:
1538:
1535:
1534:
1517:
1514:
1513:
1501:
1473:
1470:
1469:
1463:
1462:
1458:
1437:
1434:
1433:
1417:
1414:
1413:
1393:
1390:
1389:
1370:
1354:
1352:
1349:
1348:
1332:
1329:
1328:
1322:
1290:
1287:
1286:
1258:
1257:
1249:
1246:
1245:
1225:
1222:
1221:
1193:
1190:
1189:
1173:
1170:
1169:
1149:
1146:
1145:
1114:
1111:
1110:
1090:
1087:
1086:
1070:
1067:
1066:
1041:
1040:
1038:
1035:
1034:
1014:
1011:
1010:
985:
984:
970:
967:
966:
939:
936:
935:
915:
912:
911:
889:
884:
881:
880:
861:
858:
857:
851:
850:
824:
823:
821:
818:
817:
798:
795:
794:
791:
771:
714:
708:
704:
692:
688:
679:
674:
661:
657:
643:
637:
622:
618:
612:
608:
604:
600:
591:
587:
574:
572:
569:
568:
537:
534:
533:
516:
511:
498:
493:
480:
475:
469:
466:
465:
448:
444:
442:
439:
438:
417:
412:
406:
403:
402:
352:
348:
339:
335:
326:
321:
312:
309:
308:
266:
262:
257:
254:
253:
218:
215:
214:
192:
184:
134:
131:
130:
126:additive groups
119:quotient groups
83:
80:
79:
63:
55:
52:
51:
36:profinite group
28:
23:
22:
15:
12:
11:
5:
4573:
4563:
4562:
4557:
4543:
4542:
4498:
4485:
4445:
4425:
4413:(4): 475–479,
4397:
4376:
4354:(1): 239–273,
4350:, 2nd series,
4343:
4321:(1): 171–238,
4317:, 2nd series,
4310:
4296:
4276:
4275:
4257:
4250:
4222:
4213:
4204:
4192:
4183:
4140:Ann. Math. (2)
4130:
4127:on 2018-12-26.
4098:
4076:
4061:
4043:
4021:
4020:
4018:
4015:
4014:
4013:
4004:
3995:
3990:
3977:
3972:
3965:
3962:
3950:
3946:
3942:
3939:
3936:
3932:
3926:
3922:
3917:
3912:
3889:
3884:
3860:
3856:
3835:
3812:
3790:
3786:
3778:
3775:
3772:
3768:
3762:
3759:
3739:
3717:
3713:
3708:
3704:
3701:
3698:
3693:
3689:
3684:
3680:
3677:
3674:
3671:
3651:
3646:
3642:
3639:
3636:
3630:
3627:
3607:
3604:
3594:
3576:
3564:
3561:
3541:
3540:
3528:
3508:
3499:-subgroups of
3488:
3468:
3457:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3398:
3376:
3373:
3370:
3367:
3344:
3341:
3338:
3318:
3305:if it has the
3304:
3294:
3291:
3287:torsion groups
3265:locally finite
3252:
3220:
3206:
3196:
3193:
3192:
3191:
3189:
3175:
3153:
3150:
3146:
3125:
3105:
3102:
3082:
3062:
3051:
3039:
3019:
2999:
2979:
2959:
2939:
2936:
2933:
2930:
2927:
2917:
2912:
2894:
2862:
2855:
2844:
2841:
2813:
2810:
2790:
2787:
2763:
2752:
2736:
2732:
2728:
2705:
2702:
2682:
2663:
2646:
2643:
2642:
2641:
2634:
2611:
2591:
2571:
2568:
2544:
2541:
2517:
2497:
2477:
2474:
2464:
2460:
2451:Wolfgang Krull
2446:Krull topology
2432:
2429:
2425:
2421:
2418:
2415:
2412:
2392:
2387:
2383:
2379:
2374:
2370:
2349:
2346:
2342:
2336:
2332:
2328:
2325:
2322:
2319:
2316:
2313:
2309:
2303:
2299:
2295:
2292:
2289:
2279:
2265:
2261:
2257:
2237:
2217:
2214:
2211:
2207:
2203:
2200:
2197:
2194:
2174:
2154:
2130:
2127:
2123:
2119:
2116:
2113:
2110:
2107:
2104:
2080:
2076:
2072:
2053:
2044:and it is the
2033:
2028:
2023:
2001:
1998:
1993:
1988:
1967:
1963:
1959:
1954:
1950:
1946:
1942:
1938:
1933:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1867:
1863:
1859:
1854:
1833:
1829:
1805:
1801:
1783:
1772:
1767:
1762:
1740:
1720:
1717:
1714:
1711:
1690:
1684:
1680:
1675:
1670:
1666:
1662:
1656:
1652:
1647:
1642:
1617:
1596:
1590:
1586:
1581:
1576:
1549:
1544:
1532:-adic integers
1521:
1509:
1500:
1497:
1494:
1480:
1477:
1467:
1457:
1454:
1441:
1421:
1397:
1377:
1373:
1369:
1366:
1361:
1358:
1336:
1321:
1318:
1306:
1303:
1300:
1297:
1294:
1274:
1271:
1265:
1262:
1256:
1253:
1229:
1209:
1206:
1203:
1200:
1197:
1177:
1153:
1130:
1127:
1124:
1121:
1118:
1094:
1074:
1054:
1048:
1045:
1018:
998:
992:
989:
983:
980:
977:
974:
943:
919:
899:
896:
892:
888:
868:
865:
855:
837:
831:
828:
805:
802:
790:
787:
770:
767:
765:construction.
731:
727:
724:
721:
711:
707:
703:
700:
695:
691:
687:
682:
677:
673:
669:
664:
660:
652:
649:
646:
642:
636:
631:
628:
625:
621:
615:
611:
607:
603:
599:
594:
590:
586:
581:
578:
556:
553:
550:
547:
544:
541:
519:
514:
510:
506:
501:
496:
492:
488:
483:
478:
474:
451:
447:
420:
415:
411:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
355:
351:
347:
342:
338:
334:
329:
324:
320:
316:
289:
286:
283:
280:
277:
274:
269:
265:
261:
250:indexed family
237:
234:
231:
228:
225:
222:
204:inverse system
191:
188:
183:
180:
149:-adic integers
138:
104:Sylow theorems
87:
66:
62:
59:
26:
18:Krull topology
9:
6:
4:
3:
2:
4572:
4561:
4558:
4556:
4553:
4552:
4550:
4539:
4535:
4531:
4527:
4522:
4517:
4513:
4509:
4508:
4503:
4499:
4496:
4492:
4488:
4486:3-540-61990-9
4482:
4478:
4474:
4470:
4464:
4460:
4456:
4452:
4448:
4442:
4438:
4434:
4430:
4426:
4421:
4416:
4412:
4408:
4407:
4402:
4398:
4394:
4387:
4386:
4381:
4377:
4372:
4367:
4362:
4357:
4353:
4349:
4344:
4339:
4334:
4329:
4324:
4320:
4316:
4311:
4307:
4303:
4299:
4293:
4289:
4285:
4280:
4279:
4271:
4267:
4261:
4253:
4247:
4243:
4239:
4235:
4234:
4226:
4217:
4208:
4199:
4197:
4187:
4179:
4175:
4171:
4167:
4163:
4159:
4154:
4149:
4145:
4141:
4134:
4123:
4119:
4112:
4105:
4103:
4094:
4087:
4080:
4072:
4068:
4064:
4062:9780198500827
4058:
4054:
4047:
4038:
4033:
4026:
4022:
4008:
4005:
3999:
3996:
3994:
3991:
3985:
3983:
3978:
3976:
3973:
3971:
3968:
3967:
3961:
3948:
3940:
3937:
3934:
3924:
3920:
3915:
3887:
3858:
3854:
3833:
3826:
3825:prime numbers
3810:
3788:
3784:
3776:
3773:
3770:
3760:
3757:
3737:
3728:
3715:
3711:
3702:
3699:
3696:
3691:
3687:
3682:
3678:
3672:
3649:
3637:
3628:
3625:
3605:
3602:
3588:
3574:
3560:
3558:
3554:
3550:
3546:
3526:
3519:are free pro-
3506:
3486:
3466:
3458:
3444:
3441:
3438:
3432:
3426:
3423:
3416:
3412:
3411:
3410:
3396:
3387:
3374:
3371:
3365:
3358:
3342:
3336:
3316:
3308:
3298:
3290:
3288:
3284:
3280:
3275:
3273:
3270:
3266:
3250:
3242:
3238:
3234:
3218:
3210:
3200:
3187:
3173:
3151:
3148:
3144:
3123:
3103:
3100:
3080:
3060:
3052:
3037:
3017:
2997:
2977:
2957:
2937:
2931:
2928:
2925:
2915:
2913:
2910:
2906:
2892:
2882:
2878:
2875:
2871:
2867:
2863:
2860:
2856:
2842:
2839:
2832:functions on
2831:
2827:
2826:probabilities
2811:
2808:
2788:
2785:
2777:
2761:
2753:
2750:
2734:
2730:
2726:
2719:
2703:
2700:
2680:
2672:
2668:
2664:
2661:
2657:
2653:
2649:
2648:
2640:is profinite.
2639:
2635:
2632:
2628:
2624:
2620:
2616:
2612:
2609:
2605:
2589:
2569:
2566:
2558:
2542:
2539:
2531:
2530:finite groups
2515:
2495:
2475:
2472:
2462:
2458:
2456:
2452:
2448:
2447:
2427:
2423:
2419:
2413:
2410:
2390:
2385:
2381:
2377:
2372:
2368:
2344:
2340:
2334:
2330:
2323:
2320:
2311:
2307:
2301:
2297:
2290:
2287:
2277:
2263:
2259:
2255:
2235:
2215:
2209:
2205:
2201:
2195:
2192:
2172:
2152:
2144:
2125:
2121:
2117:
2111:
2108:
2105:
2102:
2094:
2078:
2074:
2070:
2062:
2058:
2057:Galois theory
2054:
2051:
2047:
2031:
2026:
1999:
1996:
1986:
1961:
1957:
1940:
1936:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1861:
1857:
1831:
1803:
1788:
1785:The group of
1784:
1770:
1765:
1738:
1718:
1715:
1712:
1709:
1682:
1678:
1673:
1654:
1650:
1645:
1631:
1615:
1588:
1584:
1579:
1565:
1547:
1533:
1519:
1511:The group of
1510:
1507:
1503:
1502:
1496:
1492:
1478:
1475:
1461:
1453:
1439:
1419:
1411:
1395:
1375:
1371:
1367:
1364:
1359:
1356:
1334:
1325:
1317:
1304:
1301:
1298:
1295:
1292:
1272:
1263:
1260:
1254:
1251:
1243:
1227:
1207:
1201:
1198:
1195:
1175:
1167:
1151:
1142:
1128:
1125:
1122:
1119:
1116:
1108:
1092:
1072:
1052:
1046:
1043:
1032:
1016:
996:
990:
987:
978:
975:
972:
963:
961:
957:
941:
933:
917:
897:
894:
890:
886:
866:
863:
849:
835:
829:
826:
803:
800:
786:
784:
780:
776:
766:
764:
760:
756:
751:
749:
746:
729:
725:
722:
719:
709:
705:
701:
693:
689:
680:
675:
671:
667:
662:
658:
650:
647:
644:
634:
629:
626:
623:
613:
609:
601:
597:
592:
588:
584:
579:
576:
554:
551:
548:
545:
542:
539:
517:
512:
508:
504:
499:
494:
490:
486:
481:
476:
472:
449:
445:
436:
418:
413:
409:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
353:
349:
340:
336:
332:
327:
322:
318:
307:
306:homomorphisms
303:
287:
281:
278:
275:
272:
267:
263:
251:
235:
229:
226:
223:
213:
209:
205:
201:
200:inverse limit
197:
187:
179:
177:
173:
169:
165:
160:
158:
154:
153:Galois groups
150:
136:
127:
122:
120:
116:
112:
107:
105:
101:
85:
60:
57:
47:
45:
44:finite groups
41:
37:
33:
19:
4511:
4505:
4472:
4432:
4410:
4404:
4384:
4351:
4347:
4318:
4314:
4283:
4270:MathOverflow
4269:
4260:
4232:
4225:
4216:
4207:
4186:
4153:math/0604399
4143:
4139:
4133:
4122:the original
4117:
4092:
4079:
4052:
4046:
4037:math/0703885
4025:
3981:
3729:
3618:that is, if
3566:
3542:
3388:
3296:
3277:By applying
3276:
3233:direct limit
3198:
2776:Haar measure
2718:factor group
2555:This is the
2457:showed that
2444:
2050:finite field
1459:
1326:
1323:
1143:
964:
792:
772:
752:
435:identity map
212:directed set
193:
185:
161:
123:
108:
48:
35:
29:
4393:Oberwolfach
3355:there is a
2610:of a field.
1493:reconstruct
1320:Equivalence
783:Stone space
759:categorical
32:mathematics
4549:Categories
4538:0281.20031
4495:0902.12004
4463:0812.12002
4306:1145.12001
4178:1126.20018
4017:References
3479:the Sylow
3301:projective
3263:is called
3190:structure.
2631:CW complex
1464:surjective
1242:continuous
954:of finite
401:such that
196:isomorphic
182:Definition
115:surjective
3941:∈
3774:∈
3767:∏
3761:≅
3703:∈
3688:σ
3676:⟩
3673:σ
3670:⟨
3645:¯
3641:⟩
3638:σ
3635:⟨
3603:σ
3591:procyclic
3439:≤
3427:
3369:→
3340:→
3267:if every
3241:ind-group
3188:algebraic
3174:ι
3149:−
3145:ι
3124:ι
3101:ι
2935:→
2926:φ
2870:Dan Segal
2830:integrate
2716:then the
2414:
2378:⊆
2324:
2318:→
2291:
2196:
2112:
1949:→
1911:…
1804:^
1713:≥
1665:→
1564:procyclic
1491:and then
1365:
1360:←
1302:η
1270:→
1264:^
1205:→
1152:η
1117:∩
1073:η
1047:^
991:^
982:→
973:η
830:^
723:≤
648:∈
641:∏
635:∈
627:∈
585:
580:←
549:≤
543:≤
532:whenever
487:∘
383:≤
371:∈
359:∣
346:→
279:∈
230:≤
61:∈
4471:(1997),
4431:(1994),
4382:(2003),
4170:15670650
4071:40658188
3964:See also
3539:-groups.
3272:subgroup
3053:Suppose
1499:Examples
745:relative
208:discrete
151:and the
102:and the
4530:2039560
4455:1324577
3357:section
3283:abelian
2652:product
2048:of any
1109:(i.e.,
775:compact
433:is the
198:to the
164:compact
4536:
4528:
4493:
4483:
4461:
4453:
4443:
4304:
4294:
4248:
4176:
4168:
4069:
4059:
3803:where
3235:of an
2905:groups
2828:, and
2667:closed
2665:Every
2650:Every
2621:. The
2465:field
2449:after
2278:finite
2228:where
1879:where
1608:where
1388:where
1220:where
910:where
777:, and
202:of an
4526:JSTOR
4389:(PDF)
4356:arXiv
4323:arXiv
4166:S2CID
4148:arXiv
4125:(PDF)
4114:(PDF)
4089:(PDF)
4032:arXiv
3984:group
3547:of a
3116:Then
2874:dense
2859:index
2673:. If
2660:exact
2459:every
2276:is a
2091:is a
1285:with
1031:dense
956:index
757:. In
38:is a
4481:ISBN
4441:ISBN
4292:ISBN
4246:ISBN
4067:OCLC
4057:ISBN
3980:Pro-
3846:and
3555:and
3413:the
3209:dual
3073:and
2970:and
2885:pro-
2868:and
2613:The
2463:some
2055:The
1979:for
1702:for
848:the
166:and
34:, a
4534:Zbl
4516:doi
4491:Zbl
4466:.
4459:Zbl
4415:doi
4366:doi
4352:165
4333:doi
4319:165
4302:Zbl
4238:doi
4174:Zbl
4158:doi
4144:165
3902:or
3587:is
2916:any
2778:on
2625:of
2411:Gal
2321:Gal
2288:Gal
2193:Gal
2145:of
2109:Gal
2059:of
1412:of
1357:lim
1105:is
1033:in
934:in
856:of
577:lim
437:on
248:an
206:of
128:of
30:In
4551::
4532:,
4524:,
4512:42
4510:,
4489:,
4479:,
4457:,
4451:MR
4449:,
4439:,
4411:38
4409:,
4364:,
4331:,
4300:.
4290:.
4268:.
4244:.
4195:^
4172:.
4164:.
4156:.
4142:.
4116:.
4101:^
4091:.
4065:.
3559:.
3424:cd
3289:.
2453:.
785:.
750:.
178:.
159:.
106:.
46:.
4541:.
4518::
4417::
4396:.
4375:.
4368::
4358::
4342:.
4335::
4325::
4308:.
4272:.
4254:.
4240::
4180:.
4160::
4150::
4095:.
4073:.
4040:.
4034::
3982:p
3949:.
3945:N
3938:n
3935:,
3931:Z
3925:n
3921:p
3916:/
3911:Z
3888:p
3883:Z
3859:p
3855:G
3834:S
3811:p
3789:p
3785:G
3777:S
3771:p
3758:G
3738:G
3716:.
3712:}
3707:Z
3700:n
3697::
3692:n
3683:{
3679:=
3650:,
3629:=
3626:G
3606:;
3575:G
3527:p
3507:G
3487:p
3467:p
3445:;
3442:1
3436:)
3433:G
3430:(
3397:G
3375:.
3372:H
3366:G
3343:G
3337:H
3317:G
3251:G
3219:G
3152:1
3104:.
3081:H
3061:G
3038:G
3018:H
2998:G
2978:H
2958:G
2938:H
2932:G
2929::
2911:.
2893:p
2861:.
2843:.
2840:G
2812:,
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