# True/False: A permutation of order 2 in $S_n$ must be odd.

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.

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