On w-maximal groups

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 $\{w(g_1,...,g_n{)}^{\pm 1}|g_i\in G,1\le i\le n\}$ 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.

15 pages