A Nilpotency Criterion for Some Verbal subgroups

Abstract

The word w = [x i1 , x i2 , . . . , x i k ] is a simple commutator word if k ≥ 2, i 1 = i 2 and i j ∈ {1, . . . , m}, for some m > 1. For a finite group G, we prove that if i 1 = i j for every j = 1, then the verbal subgroup corresponding to w is nilpotent if and only if |ab| = |a||b| for any w-values a, b ∈ G of coprime orders. We also extend the result to a residually finite group G, provided that the set of all w-values in G is finite.

Coprime commutators in finite groups

“…Proposition 18 ( [9]). Every finite minimal simple group G contains a subgroup H = A ⋊ T where A is an elementary abelian 2-group and T is a subgroup of odd order such that C A (T ) = 1.…”

“…Also in the case of commutator words Question 3 has negative answer since one can consider the alternating group A 5 of degree 5 and the word [x, y 10 , y 10 , y 10 ] all of whose nontrivial values in A 5 have order 2 (see [9]).…”

$p$-nilpotency criteria for some verbal subgroups

“…The work of Baumslag and Wiegold has been extended to the realm of group-words in a series of papers [3,4,5,8,12]. In a similar matter, the aim of this work is to generalize Theorem A obtaining a verbal version.…”

“…Even in the case of commutator words we can find counterexamples. Indeed, if G = Alt(5) is the alternating group of degree 5 and w is the word considered in [12,Example 4.2], then G w consists of the identity and all products of two transpositions. In particular if p ∈ {2, 3, 5} then G satisfies P (w, p), but w(G) = G is a simple group and therefore not p-nilpotent.…”

On nilpotency of higher commutator subgroups of a finite soluble group

“…We end this short introduction by mentioning that there are several other recent results related to the theorem of Baumslag and Wiegold (see in particular [4,7,9,[11][12][13]).…”