On periodic groups with small orders of elements

Mazurov, V. D.; Mamontov, A. S.

doi:10.1007/s11202-009-0036-1

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…The theorem gives an affirmative answer to question 16.56 in [2]. Together with results in [3][4][5][6][7][8][9][10][11][12], the theorem implies the following:…”

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confidence: 58%

“…By Thompson's theorem, the kernel the Frobenius group U is nilpotent, in which case U contains an element of order 12, which is impossible. Thus G/H is not a 5-group and, therefore, is locally finite [4,8,11]. In view of [12], H is also a locally finite group, and G is locally finite by virtue of Schmidt's theorem (see [16,Thm.…”

Section: The Notation and Preliminary Resultsmentioning

confidence: 99%

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Groups Whose Element Orders do not Exceed 6

et al. 2014

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It is proved that a periodic group whose element orders do not exceed 6 either is a locally finite or is group of exponent 5.The spectrum of a group G is the set ω(G) of its element orders. The objective of the present paper is to prove THEOREM. If the spectrum of a group G is equal to {1, 2, 3, 4, 5, 6}, then G is locally finite and satisfies one of the following:(where C is isomorphic to SL 2 (3) or to a group x, y | x 3 = y 4 = 1, x y = x −1 , and C acts freely on N ;(2) T = O 2 (G) is a nontrivial elementary Abelian group and G/T is isomorphic to A 5 ;(3) G is isomorphic to S 5 or S 6 . Hereinafter, O p (G), where p is a prime, denotes the greatest normal p-subgroup of G; A n and S n stand for, respectively, the alternating group of degree n and the symmetric group of degree n. A group A freely acts on a group B if A acts on B, B is nontrivial, and b a = b if a and b are nontrivial elements of A and B, respectively.

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“…The theorem gives an affirmative answer to question 16.56 in [2]. Together with results in [3][4][5][6][7][8][9][10][11][12], the theorem implies the following:…”

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confidence: 58%

Section: The Notation and Preliminary Resultsmentioning

confidence: 99%

Groups Whose Element Orders do not Exceed 6

et al. 2014

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“…, then G is also locally finite (see [7,10,15,17,18] respectively). The same is true for a group G such that ω(G) = {1, 2, 3, 5, 9, 15} or ω(G) = {2, 3} ∪ ω, where every element of ω is either coprime to 6, or equals 9 [25].…”

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confidence: 99%