Cyclic extensions of free pro-$p$ groups and $p$-adic modules

Abstract

Abstract. We prove a pro-p version of the classical decomposition of a Z p -torsion free Z p C p -module into indecomposable modules. We also describe some pro-p Z p C p nmodules obtained from a semidirect product of a free pro-p group F and a cyclic group C p n of automorphisms by factoring out the (closed) commutator subgroup [F, F ].

Infinitely generated virtually free pro‐p groups and p‐adic representations

“…Thus we have the following commutative diagram: Since is a ‐invariant free factor of , its abelianization is a pure ‐submodule of (that is, a ‐direct summand). By [, Theorem B] , where is a free ‐module and has no non‐zero elements fixed by . Since is free it is a direct summand of (see [, Proposition 3.6.4]; the proof over works mutatis mutandis) and so by Proposition (ii) can be assumed to be contained in .…”