Bounding the Exponent of a Verbal Subgroup

2013/01/17 by Detomi, Eloisa, Morigi, Marta, Shumyatsky, Pavel
#20F10 #20F14 #20F45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1301.4093

Abstract

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 [xⁿ,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).

Related

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• A nilpotency criterion for some verbal subgroups
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• Words of Engel type are concise in residually finite groups