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I am trying to prove the validity of the statement: A permutation of order 2 in $S_n$ must be odd.

I know that $S_n$ is the set of all permutations and that (with $k \geq 2$) any k-cycle is a product of k-1 transpositions. Likewise, if k is even, it's an odd permutation and if k is odd, it's an even permutation. Hence, if $\sigma$ is an element in $S_n$ then $\sigma$ is a product of 2-1 transpositions. Therefore, $\sigma$ is an odd permutation. Thus, my conjecture is that this is a true statement. I would appreciate any feedback.

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$(1\,2)(3\,4)\in S_4$ is of order $2$ and is even

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