Endomorphisms of profinite groups

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 $\varphi$ we obtain a semidirect decomposition of $G$ into a ‘contracting’ normal subgroup and a complement on which $\varphi$ 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 $p$.