[FOM] Weak choice needed

[joeshipman at aol.com]

I can prove “every group of order >2 has a nontrivial automorphism” with the Boolean Prime Ideal Theorem, a weak form of Choice. Is this necessary? Is there a yet weaker form of Choice that suffices?

It comes down to whether an infinite abelian group G where every element has order 2 is isomorphic to the direct product of a subgroup of order 2 with the quotient group of cosets of that subgroup. Easy to get by expressing G as a vector space over F2 and using the Boolean Prime Ideal Theorem to get a basis; but I don’t need a full basis, I just need an isomorphism between G and H x (G/H) for some 2-element subgroup H. Is that strictly weaker?

— JS

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