Question about AC

[Joe Shipman]

But G is infinite so you still need choice to get an automorphism, as suggested here:

https://math.stackexchange.com/questions/1105081/why-any-short-exact-sequence-of-vector-spaces-may-be-seen-as-a-direct-sum

Sent from my iPhone

> On May 13, 2020, at 4:58 PM, Ingo Blechschmidt <iblech at speicherleck.de> wrote:
> 
> 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
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