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are considered: Does specifying an exponent force finiteness? The existence of infinite, finitely generated periodic groups as in the previous paragraph shows that the answer is "no" for an arbitrary exponent. Though much more is known about which exponents can occur for infinite finitely generated
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and is therefore inadmissible: first order logic permits quantifiers over one type and cannot capture properties or subsets of that type. It is also not possible to get around this infinite disjunction by using an infinite set of axioms: the
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Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the
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However, in first-order logic we may not form infinitely long disjunctions. Indeed, we shall later show that there is no set of first-order formulas whose models are precisely the periodic groups.
86:. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich (see
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399:, An infinite group with subgroups of prime orders, Math. USSR Izv. 16 (1981), 279–289; translation of Izvestia Akad. Nauk SSSR Ser. Matem. 44 (1980), 309–321
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Burnside's problem is a classical question that deals with the relationship between periodic groups and
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262:{\displaystyle \forall x,{\big (}(x=e)\lor (x\circ x=e)\lor ((x\circ x)\circ x=e)\lor \cdots {\big )},}
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An interesting property of periodic groups is that the definition cannot be formalized in terms of
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that every finite group is periodic and it has an exponent that divides its order.
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implies that no set of first-order formulae can characterize the periodic groups.
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Degrees of growth of finitely generated groups and the theory of invariant means
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19:"Periodic group" redirects here. For groups in the chemical periodic table, see
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is an abelian group in which every element has finite order. A
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Finite automata and the
Burnside problem for periodic groups
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groups there are still some for which the problem is open.
415:(2. ed., 4. pr. ed.). New York : Springer. pp.
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that consists of all elements that have finite order. A
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347:On nil-algebras and finitely approximable p-groups
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409:Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994).
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16:Group in which each element has finite order
82:. Another example is the direct sum of all
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381:On Burnside's problem on periodic groups
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58:of such a group, if it exists, is the
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452:, Izv. Akad. Nauk SSSR Ser. Mat.
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456:(1984), 939–985 (Russian).
310:torsion-free abelian group
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383:, Functional Anal. Appl.
366:, (Russian) Mat. Zametki
120:finitely generated groups
89:Golod–Shafarevich theorem
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21:Group (periodic table)
387:(1980), no. 1, 41–43.
306:torsion abelian group
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98:Tarski monster groups
60:least common multiple
470:Properties of groups
327:Jordan–Schur theorem
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300:is the subgroup of
279:compactness theorem
448:R. I. Grigorchuk,
412:Mathematical logic
379:R. I. Grigorchuk,
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134:Mathematical logic
110:Burnside's problem
104:Burnside's problem
67:Lagrange's theorem
426:978-0-387-94258-2
397:A. Yu. Olshanskii
322:Torsion (algebra)
140:first-order logic
73:Infinite examples
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52:finite order
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28:group theory
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274:disjunction
32:mathematics
333:References
247:⋯
244:∨
229:∘
220:∘
208:∨
193:∘
184:∨
153:∀
464:Category
316:See also
94:automata
56:exponent
432:18 July
48:element
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293:of an
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434:2012
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