On w-maximal groups

TL;DR

This paper investigates the properties and classifications of finite groups that are maximal with respect to a specific verbal subgroup defined by a word, focusing on their structure and hereditary properties.

Contribution

It introduces the concepts of w-maximal and hereditarily w-maximal groups and studies their structural properties and classifications.

Findings

Characterization of w-maximal groups

Conditions for hereditary w-maximality

Examples of such groups and their properties

Abstract

Let $w = w(x_1,..., x_n)$ be a word, i.e. an element of the free group $F =<x_1,...,x_n>$ on $n$ generators $x_1,..., x_n$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{w (g_1,...,g_n)^{\pm 1} | g_i \in G, 1\leq i\leq n \}$ of all $w$-values in $G$. We say that a (finite) group $G$ is $w$-maximal if $|G:w(G)|> |H:w(H)|$ for all proper subgroups $H$ of $G$ and that $G$ is hereditarily $w$-maximal if every subgroup of $G$ is $w$-maximal. In this text we study $w$-maximal and hereditarily $w$-maximal (finite) groups.