TITLE: Group theory and direct product
QUESTION [5 upvotes]: Let $G$ be a finite group such that $x^2=e$ for any $x \in G$ (that is: every element other than identity is of order $2$). Then $G$ is isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2\times\ldots\times\mathbb{Z}_2$.

REPLY [2 votes]: The fact that $x^2=e$ implies that $G$ is abelian since $(ab)^2=e$ so $abab=e$ hence $ba=a^{-1}b^{-1}=ab$ (because $a,b$ of order 2). Moreover you have that 2 divides the order of $G$. The result follows from Fundamental theorem of finitely generated abelian groups.