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I'm trying to look for a group such that the number of elements $x$ with $x^2 = e$ is finite, odd and not 1. That is, a group with a finite even number of order 2 elements.

I can't find an example of such group and I can't figure out how to prove if it can exist or not.

I know that this group has got to be non-abelian. If the group was abelian, the set $B$ of order 2 elements would also be a group and the subgroup generated by any of its non-trivial elements has order 2, so the set $B$ has an even number of elements (if it is finite).

However, for a non-abelian group, $B$ isn't necessarily a group (For example, the transpositions of $S_n$ have order 2 and generate $S_n$. If $B$ was always group, this would mean that $B=S_n$, which is not the case for $n > 2$), so I can't think of any way to prove this for non-abelian groups.

dino
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    I have to say it's not obvious to me how the duplicate link answers this question. There is no such finite group, which follows from this and this (here it's basically a corollary of the usual proof of this special case of Cauchy's theorem). There's also no such infinite group, which follows from this. – Izaak van Dongen Nov 08 '23 at 20:48

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