 Solution

Suppose $\sigma$ has order of a group element two. On a cycle of length $k$, ${\sigma }^2$ moves each entry two places. Since ${\sigma }^2=e$, every such $k$ divides two, so $k=1$ or $k=2$. After omitting the fixed points, the disjoint cycle decomposition is therefore a product of disjoint permutation cycles of length two, which are transpositions. There is at least one, because $\sigma \ne e$.