Question about AC

[Ingo Blechschmidt]

Dear Joe,

On Tue 12 May 2020 01:25:20 PM GMT, Joe Shipman wrote:
> Let G be an abelian group and H be a finite subgroup of G.
> Is some form of AC necessary to prove that there exists a group K such that G is isomorphic to H x K ?

this statement is false in general [1], but it is true in case every
element of G is its own inverse. In this case G can be regarded as an
��₂-vector space, hence the result follows from the fact that short exact
sequences of vector spaces split.

As H is finite, hence finite-dimensional, no choice is required.
(In fact, not even the law of excluded middle is required.)

Cheers,
Ingo

[1] https://math.stackexchange.com/a/544193