TITLE: If in a group G an element $x$ has the property such that $x^2 = e$, does that entail $x = e$?
QUESTION [7 upvotes]: I can't seem to prove that $x = e$ from $x = x^{-1}$.

REPLY [3 votes]: [$x^2=e \Rightarrow x=e$] if the order of $x$ is odd. For let $n$ be the order of $x$ and write $n=2m+1$, assume $x^2=e$. Then $e=x^n=x^{2m+1}=(x^2)^m \cdot x= e^m \cdot x= x$.