TITLE: The number of elements of order $2$ in an infinite group
QUESTION [10 upvotes]: I've just proved that for $|G|$ finite, the number of elements of order $2$ is odd. But how about the case $|G|$ infinite?
(Here, there is assumption that the set of elements of order $2$ is nonempty.)

REPLY [6 votes]: Let $t$ be an element of order $2$ in $G$, and assume that the number of elements of order $2$ is finite. It is enough to prove that $\{x \in G: x^2 = 1\}$ contains an even number of elements. To do that, you can pair each $x \in C_G(t)$ with $xt$ and each $x \not\in C_G(t)$ with $txt^{-1}$.