Galois-theoretic features for 1-smooth pro-p groups

Abstract. Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation θ : G → GL1 (Z p ) such that every open subgroup H of G, together with the restriction θ |H , satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally we ask whether 1-smooth pro-p groups satisfy a “Tits’ alternative”.