Endomorphisms of profinite groups

  • Colin D. Reid

    The University of Newcastle, Callaghan, Australia


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

Cite this article

Colin D. Reid, Endomorphisms of profinite groups. Groups Geom. Dyn. 8 (2014), no. 2, pp. 553–564

DOI 10.4171/GGD/238