Dihedral trick

Statement with symbols
Let $$x$$ and $$y$$ be two distinct involutions (elements of order two) in a finite group $$G$$. Suppose $$xy$$ has order $$m$$. Then, $$\langle x, y \rangle$$ is a dihedral group of order $$2m$$, with cyclic subgroup of order $$m$$ generated by $$xy$$ and the element $$x$$ of order two conjugating $$xy$$ to its inverse.

Applications

 * Involutions are either conjugate or have an involution centralizing both of them