On groups with large verbal quotients

Abstract

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F=⟨x_1,...,x_n⟩$. 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. González-Sánchez 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 $n$.

12 pages, to appear in J. Group Theory

References in corpus (2)

• Nilpotent subgroups of class $2$ in finite groups
• The growth of abelian sections

Cited by in corpus (1)

• Probabilistic construction of some extremal $p$-groups