A short proof of the Khukhro--Makarenko theorem on large characteristic subgroups with laws

TL;DR

This paper presents a concise proof and enhancements of the Khukhro--Makarenko theorem, showing that groups nearly satisfying an outer commutator law contain a finite-index characteristic subgroup that fully satisfies the law, with an estimated index.

Contribution

It provides a shorter proof and strengthens the theorem, including an index estimate for the characteristic subgroup.

Findings

Shortened proof of the Khukhro--Makarenko theorem

Strengthened version with improved conditions

Derived estimate for the subgroup index

Abstract

We give a short proof and some strengthening of the Khukhro--Makarenko theorem that each group virtually satisfying an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained.