A focal subgroup theorem for outer commutator words

TL;DR

This paper extends the subgroup theorem to outer commutator words in finite groups, showing how Sylow p-subgroups intersect with the values of these words.

Contribution

It establishes a new subgroup theorem for outer commutator words in finite groups, generalizing previous results.

Findings

P∩w(G) is generated by P∩mth powers of w-values

The result applies to groups of order p^a m with m coprime to p

Provides a structural insight into the interaction between Sylow subgroups and outer commutator words

Abstract

Let $G$ be a finite group of order $p^am$, where $p$ is a prime and $m$ is not divisible by $p$, and let $P$ be a Sylow $p$-subgroup of $G$. If $w$ is an outer commutator word, we prove that $P\cap w(G)$ is generated by the intersection of $P$ with the set of $m$th powers of all values of $w$ in $G$