Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type

TL;DR

This paper provides a new, classification-independent proof that quasiprimitive permutation groups of simple diagonal type cannot embed into wreath products in product action, using deep results on factorisations and automorphisms.

Contribution

It introduces a novel proof method applicable to both finite and infinite groups, linking factorisations and automorphisms to group embedding properties.

Findings

Quasiprimitive groups of simple diagonal type cannot embed into wreath products in product action.

Factorisations involving subdirect subgroups are governed by uniform automorphisms.

The proof is independent of the finite simple group classification.

Abstract

We present a new proof, which is independent of the finite simple group classification and applies also to infinite groups, that quasiprimitive permutation groups of simple diagonal type cannot be embedded into wreath products in product action. The proof uses several deep results that concern factorisations of direct products involving subdirect subgroups. We find that such factorisations are controlled by the existence of uniform automorphisms.