We obtain some general restrictions on the continuous endomorphisms of a profinite group under the assumption that has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if is finitely generated). In particular, given such a group and a continuous endomorphism we obtain a semidirect decomposition of into a `contracting' normal subgroup and a complement on which induces an automorphism; both the normal subgroup and the complement are closed. If is isomorphic to a proper open subgroup of itself, we show that has an infinite abelian normal pro- subgroup for some prime .
Cite this article
Colin D. Reid, Endomorphisms of profinite groups. Groups Geom. Dyn. 8 (2014), no. 2, pp. 553–564DOI 10.4171/GGD/238