On the rank of a verbal subgroup of a finite group

TL;DR

This paper establishes bounds on the rank of verbal subgroups generated by multilinear commutator words in finite groups, based on the ranks of certain subgroups, with improved bounds for soluble groups.

Contribution

It provides new bounds on the rank of verbal subgroups in finite groups, depending on subgroup ranks and group solubility, advancing understanding of group structure.

Findings

Bound on rank of $w(G)$ in finite groups with metanilpotent subgroup conditions.

Improved bound for soluble groups with nilpotent subgroup conditions.

Explicit relation between subgroup ranks and verbal subgroup rank.

Abstract

We show that if $w$ is a multilinear commutator word and $G$ a finite group in which every metanilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of the verbal subgroup $w(G)$ is bounded in terms of $r$ and $w$ only. In the case where $G$ is soluble we obtain a better result -- if $G$ is a finite soluble group in which every nilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of $w(G)$ is at most $r+1$.