I have the following problem:

Let $H$ be the subgroup of $S\_n$, where $S\_n$ denotes the group of injective functions mapping a set to itself, and the order of $H$ is odd. I need to prove that $H$ is a subgroup of $A\_n$ where $A\_n$ is an alternating group of degree n.

My idea: If the order of $H$ is odd, then it means that the least common multiple of length of the disjoint cycles that make up $H$ is odd, which implies that there are only cycles of odd length. I think all cycles can be written as a multiple of 2-cycles, thus, I can conclude that $H$ is a subgroup of the group of even permutations?

Am I right? Any help would be appreciated!

Thanks in advance!