Permutations of Symmetric Group of Order 3 Find an example, in the group $S_3$ of permutations of $\{1,2,3\}$, of elements $x,y\in S_3$ for which $x^2 = e = y^2$ but for which $(xy)^4$ $\not=$ e.

This group is isomorphic to the 6 element dihedral group. Any element including a reflection will have order two. The product of two different such elements will be a pure nonzero rotation, necessarily of order 3.
