I am reading Iwasawa's paper 'On Γ-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=Γ. Here Γ is topologically isomorphic to the additive group of 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.

Question

I am reading Iwasawa&#039;s paper &#039;On Γ-extensions of Algebraic Number Fields&#039;. 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=Γ. Here Γ is topologically isomorphic to the additive group of             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&#039;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.

Vitulloh0 · Accepted Answer

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 &quot;universal property&quot;, 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.

I am reading Iwasawa's paper 'On Γ-extensions of Algebraic Number Fields'. On Page 189, he discussed

Answered question

Answer & Explanation

New Questions in Pre-Algebra