On groups with large verbal quotients

TL;DR

This paper investigates the properties of groups with large verbal quotients, introducing new results on $w$-maximal groups and their weaker variants, with applications to finite groups containing solvable or nilpotent sections.

Contribution

It provides novel insights into $w$-maximal groups and explores the implications of verbal quotient sizes for subgroup structures in finite groups.

Findings

Finite groups with solvable sections have large solvable subgroups.

Finite groups with nilpotent sections have large nilpotent subgroups.

New bounds and properties for $w$-maximal groups are established.

Abstract

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all $w$-values in $G$. Following J. Gonz\'alez-S\'anchez and B. Klopsch, a group $G$ is $w$-maximal if $|H:w(H)| < |G:w(G)|$ for every $H<G$. In this paper we give new results on $w$-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size $n$, then it has a solvable (resp. nilpotent) subgroup of size at least $n$.