I am reading Iwasawa's paper 'On $\Gamma$-extensions of Algebraic Number Fields'. On Page 189, he discussed the following data:
Let $G$ be a compact group, $X$ a closed normal subgroup of $G$ such that $X$ is a $p$-primary compact abelian group (meaning that $X$ is an abelian pro-$p$ group) and that $G/N=\Gamma$. Here $\Gamma$ is topologically isomorphic to the additive group of $\mathbb{Z}_p$.
Iwasawa claimed that it is easy to see that the group extension $G/X$ splits (first line of Page 190), but I don't see a proof. My knowledge on topological groups is limited, so I would appreciate it very much if anyone can provide me with a proof or some references.
The main point actually is that the group Z_p is free (on one generator) in the category of (not necessarily abelian) pro-p-groups. Expressed in terms of a "universal property", this means that a free pro-p-group F_S on a set S of generators, is characterized by the following property : homomorphisms F_S → G (for any pro-p-group G) are in one-to-one correspondence with functions S → G. For a non-free group, the presence of relations would restrict the possible images of the generators under a homomorphism (Wiki). This shows readily that for any pro-p-group G which admits Z_p as a quotient, the given surjective homomorphism admits a lift (a homomorphism, not only an ensemblist section),in other words, G is a semi-direct product.