On Verbal Subgroups in Residually Finite Groups

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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“…The next lemma is Lemma 2.2 of [3]. In the case where k = 1 this is a well-known result, due to Mann [10].…”

Section: Theorem Amentioning

confidence: 75%

Bounding the exponent of a verbal subgroup

Detomi

Morigi

Shumyatsky

2013

Annali di Matematica

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“…If is a law in a finite group , then has -bounded exponent (the case is a well-known result, due to Mann [24]; see [5, lemma 2.2] for the case ). If the -Engel word , where is repeated times, is a law in a finite group , then has a normal subgroup such that the exponent of is -bounded while is nilpotent with -bounded class [3].…”

Section: Probabilistic Almost Nilpotency Of Finite Groupsmentioning

confidence: 99%

On the commuting probability for subgroups of a finite group

Detomi¹,

Shumyatsky²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let $K$ be a subgroup of a finite group $G$ . The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$ . Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$ . We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$ -bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$ . We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^{*}(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$ -bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$ , or a Sylow subgroup, etc.

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On Verbal Subgroups in Residually Finite Groups

Cited by 2 publications

References 27 publications

Bounding the exponent of a verbal subgroup

Bounding the exponent of a verbal subgroup

On the commuting probability for subgroups of a finite group

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