Large and symmetric: The Khukhro--Makarenko theorem on laws --- without laws

TL;DR

This paper generalizes the Khukhro--Makarenko theorem to broader structures, establishing new results in groups, algebras, and graphs by removing the necessity of laws, and exploring dualities and graph-theoretic analogues.

Contribution

It extends the Khukhro--Makarenko theorem to a more general setting without requiring laws, impacting various mathematical structures and answering open questions.

Findings

Generalized the Khukhro--Makarenko theorem to structures without laws

Derived new results for groups, algebras, and graphs

Provided a graph-theoretic analogue involving planarity

Abstract

We prove a generalisation of the Khukhro--Makarenko theorem on large characteristic subgroups with laws. This general fact implies new results on groups, algebras, and even graphs and other structures. Concerning groups, we obtain, e.g., a fact in a sense dual to the Khukhro--Makarenko theorem. A graph-theoretic corollary is an analogue of this theorem in which planarity plays the role of a multilinear identity. We answer also a question of Makarenko and Shumyatsky.