Extensible endomorphisms of compact groups

2023/09/22 by Chirvasitu, Alexandru
#20C25 #20G41 #20J06 #20K27 #20K30 #20K35 #20K40 #20K45 #22C05 #22D35 #22D45 #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2309.12791

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 \mathbbA 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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