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

Statement
Suppose $$G$$ is a finite group and $$x,y$$ are two involutions of $$G$$. Then, at least one of these two is true:


 * 1) $$x$$ and $$y$$ are in the same conjugacy class.
 * 2) There exists an involution $$z$$ centralizing both $$x$$ and $$y$$.

Facts used

 * 1) uses::Dihedral trick

Proof
Given: A finite group $$G$$, two involutions $$x,y$$ of $$G$$.

To prove: Either $$x$$ or $$y$$ are conjugate or there is an involution centralizing both of them.

Proof: By fact (1), $$\langle x,y \rangle$$ is a dihedral group of order $$2m$$, where $$m$$ is the order of $$xy$$. This dihedral group has $$xy$$ generating a cyclic subgroup of order $$m$$ and $$x$$ acts on this by inverting $$xy$$.

If $$m$$ is odd, then $$x,y$$ are conjugate in $$\langle x,y \rangle$$, hence in $$G$$. Otherwise, $$m$$ is even, in which case the element $$z = (xy)^{m/2}$$ is an involution centralizing both $$x$$ and $$y$$.