On endomorphisms of profinite groups

TL;DR

This paper investigates the structure of continuous endomorphisms of profinite groups with finitely many open subgroups of each index, revealing a semidirect decomposition and conditions for self-isomorphism.

Contribution

It provides new restrictions on endomorphisms of such groups and describes their structural decomposition, especially when the group is self-similar.

Findings

Semidirect decomposition of G into a contracting subgroup and an automorphism-induced complement

Existence of an infinite abelian normal pro-p subgroup if G is isomorphic to a proper open subgroup of itself

Restrictions on endomorphisms based on the group's subgroup structure

Abstract

We obtain some general restrictions on the continuous endomorphisms of a profinite group G under the assumption that G has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if G is finitely generated). In particular, given such a group G and a continuous endomorphism phi we obtain a semidirect decomposition of G into a 'contracting' normal subgroup and a complement on which phi induces an automorphism; both the normal subgroup and the complement are closed. If G is isomorphic to a proper open subgroup of itself, we show that G has an infinite abelian normal pro-p subgroup.