Multilinear commutators in residually finite groups

Shumyatsky, Pavel

doi:10.1007/s11856-011-0157-7

Cited by 6 publications

(23 citation statements)

References 20 publications

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“…However, the next result gives a sufficient condition for a quotient to be locally graded. Shumyatsky in [15] has showed that if w is a multilinear commutator and G is a residually finite group in which for any product of at most 896 w-values x there exists a positive integer q = q(x) dividing a fixed positive integer m such that x q = 1, then the verbal subgroup w(G) is locally finite. Next, we extend this result to the class of locally graded groups.…”

Section: Preliminariesmentioning

confidence: 99%

“…We need the following result, due to Shumyatsky [15]. In particular, every w-value in G has finite order.…”

Section: Preliminariesmentioning

confidence: 99%

“…For more details concerning groups with n-Engel word-values see [2,3,13,16]. In [15] Shumyatsky has showed that if w is a multilinear commutator and G is a residually finite group in which every product of at most 896 w-values has order dividing n, then w(G) is locally finite. In the present paper we establish the following related result.…”

Section: Introductionmentioning

confidence: 99%

“…Then w(G) is locally finite.Proof. By Lemma 4.2 of[15] we have a positive integer k such that every δ k -value in G is a w-value. Therefore G (k) is locally finite by Lemma 2.3.…”

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confidence: 99%

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On Residually Finite Groups with Engel-like Conditions

Bastos

2016

Communications in Algebra

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Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) m such that x q is n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that x q is n-Engel. Then w(G) is locally virtually nilpotent.2010 Mathematics Subject Classification. 20F45; 20E26.

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Section: Preliminariesmentioning

confidence: 99%

“…We need the following result, due to Shumyatsky [15]. In particular, every w-value in G has finite order.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Then w(G) is locally finite.Proof. By Lemma 4.2 of[15] we have a positive integer k such that every δ k -value in G is a w-value. Therefore G (k) is locally finite by Lemma 2.3.…”

mentioning

confidence: 99%

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On Residually Finite Groups with Engel-like Conditions

Bastos

2016

Communications in Algebra

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“…The problematic Lemma 2.5 of [13] was later used in [14] and [4]. In [14] it was used in the proof of the following result ( [14,Proposition 3.4…”

Section: Correcting An Error In An Earlier Papermentioning

confidence: 99%

Nonsoluble and non-p-soluble length of finite groups

Khukhro

Shumyatsky

2015

Isr. J. Math.

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Abstract. Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length λ(G) as the minimum number of nonsoluble factors in a series of this kind. Upper bounds for λ(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that λ(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that λ(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length λ p (G) is introduced, and it is proved that λ p (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-psoluble length λ p (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author's paper "Multilinear commutators in residually finite groups", Israel J. Math. 189 (2012), 207-224.

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Bounding the exponent of a verbal subgroup

Detomi

Morigi

Shumyatsky

2013

Annali di Matematica

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We deal with the following conjecture. If w is a group word and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only. We show that this is true in the case where w is either the nth Engel word or the word [x n , y1, y2, . . . , y k ] (Theorem A). Further, we show that for any positive integer e there exists a number k = k(e) such that if w is a word and G is a finite group in which any nilpotent subgroup generated by products of k values of the word w has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only (Theorem B).

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Multilinear commutators in residually finite groups

Abstract: Abstract. The following result is proved. Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.

Cited by 6 publications

References 20 publications

On Residually Finite Groups with Engel-like Conditions

On Residually Finite Groups with Engel-like Conditions

Nonsoluble and non-p-soluble length of finite groups

Bounding the exponent of a verbal subgroup

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