Groups, Geometry, and Dynamics

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Volume 8, Issue 2, 2014, pp. 553–564
DOI: 10.4171/GGD/238

Published online: 2014-07-08

Endomorphisms of profinite groups

Colin D. Reid[1]

(1) The University of Newcastle, Callaghan, Australia

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 for some prime $p$.

Keywords: Profinite groups, endomorphisms of groups

Reid Colin: Endomorphisms of profinite groups. Groups Geom. Dyn. 8 (2014), 553-564. doi: 10.4171/GGD/238