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We know that an involution is any permutation $\pi$ such that $\pi^2=id$.

We also know that the number of involutions of $S_n$ will be the number of permutations $\pi\in S_n$ such that $\pi$ has order $2.$

But can you prove that this number will always be odd?

Shaun
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    The number of involutions in any finite group of even order is odd. It is probably easier to prove that! – Derek Holt Mar 23 '24 at 17:32
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    Your two definitions of involution are inconsistent. Note that ${\pi \in S_2: \pi^2 = id} = S_2$ has size $2$... – Jair Taylor Mar 23 '24 at 18:38

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