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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.
97:. 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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410:, 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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273:{\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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30:"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.
426:(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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358:On nil-algebras and finitely approximable p-groups
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420:Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994).
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27:Group in which each element has finite order
93:. Another example is the direct sum of all
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69:of such a group, if it exists, is the
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467:(1984), 939–985 (Russian).
321:torsion-free abelian group
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394:, Functional Anal. Appl.
377:, (Russian) Mat. Zametki
131:finitely generated groups
100:Golod–Shafarevich theorem
18:Exponent (group theory)
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32:Group (periodic table)
398:(1980), no. 1, 41–43.
317:torsion abelian group
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109:Tarski monster groups
71:least common multiple
481:Properties of groups
338:Jordan–Schur theorem
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311:is the subgroup of
290:compactness theorem
459:R. I. Grigorchuk,
423:Mathematical logic
390:R. I. Grigorchuk,
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145:Mathematical logic
121:Burnside's problem
115:Burnside's problem
78:Lagrange's theorem
437:978-0-387-94258-2
408:A. Yu. Olshanskii
333:Torsion (algebra)
151:first-order logic
84:Infinite examples
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441:. Retrieved
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285:disjunction
43:mathematics
344:References
258:⋯
255:∨
240:∘
231:∘
219:∨
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164:∀
475:Category
327:See also
105:automata
67:exponent
443:18 July
59:element
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304:of an
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