Extensible endomorphisms of compact groups

TL;DR

This paper characterizes certain endomorphisms of compact connected groups, showing that only trivial and inner automorphisms extend to all overgroups, with special cases for abelian groups including the negative identity.

Contribution

It provides a classification of endomorphisms of compact connected groups that extend to all overgroups, extending known results from discrete to compact groups.

Findings

Endomorphisms extending to all overgroups are trivial or inner automorphisms.

For compact connected abelian groups, the negative identity also extends.

Connectedness is essential for the main results.

Abstract

We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if $\mathbb{A}$ is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include $-\mathrm{id}$ (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.