Endomorphisms of profinite groups

Colin D. Reid¹ 

¹The University of Newcastle, Callaghan, Australia

Groups, geometry, and dynamics, Tome 8 (2014) no. 2, pp. 553-564

Résumé

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 φ we obtain a semidirect decomposition of G into a ‘contracting’ normal subgroup and a complement on which φ 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.

Détail

DOI : 10.4171/ggd/238

Classification : 20-XX
Mots-clés : Profinite groups, endomorphisms of groups