Hopf property of a topologically finitely generated profinite group
= Hopf property of a topologically finitely generated profinite group
Every <topologically finitely generated group> that is profinite has the <topological Hopf property>. There are only finitely many open subgroups of each given index, and inverse image under a surjective endomorphism permutes them. The endomorphism kernel consequently lies in every open normal subgroup and is trivial.