Mathematics – Group Theory
Scientific paper
2011-12-16
Mathematics
Group Theory
8 pages
Scientific paper
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 phi we obtain a semidirect decomposition of G into a 'contracting' normal subgroup and a complement on which phi 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.
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