Permutation-extensible: a Route to Extensible Automorphisms

This article describes the use of permutation representations to obtain a partial solution to the Extensible Automorphisms Problem. It introduces a notion of permutation-extensibility and proves the equivalence of this notion with the notion of being subgroup-conjugating. 1. The problem we want to solve 1.1. Three big problems. A quick recall of three definitions: Definition. (1) An automorphism σ of a group G is termed extensible(defined) if for any embedding G ≤ H, there is an automorphism σ′ of H such that σ|G = σ. For a particular embedding G ≤ H, the group of automorphisms of G that can be extended to automorphisms of H is denoted as Extensible ( G ; H). The group of extensible automorphisms of G is thus given by: