What question did this study set out to answer?

The research aims to identify the endomorphisms of compact connected groups that can extend to compact overgroups.

March 4, 2026Open Access

Extensible Endomorphisms of Compact Groups

Key Points

The research aims to identify the endomorphisms of compact connected groups that can extend to compact overgroups.
Analyzed endomorphisms of compact connected groups.
Compared findings with existing results on discrete groups by Schupp and Pettet.
Investigated abelian groups and their unique endomorphisms in relation to compact spaces.
Identified trivial endomorphisms and inner automorphisms as extensible.
Discovered that in compact connected abelian groups, $- extrm{id}$ is also extensible alongside the identity.
Confirmed that connectedness is essential for the identified 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 A is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include -id (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.

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