Solution
= Solution
Every element satisfies $x^2=e$, including the identity. For any $x,y\in G$,
$$
e=(xy)^2=xyxy.
$$
Since $x^{-1}=x$ and $y^{-1}=y$, this implies $xy=yx$, so $G$ is abelian. It is therefore a vector space over the two-element field, with group operation as vector addition and scalar multiplication by zero or one. A basis of this finite vector space gives
$$
\boxed{G\cong C_2^r
=\underbrace{C_2\times\cdots\times C_2}_{r\text{ factors}}}
$$
for some $r\geq0$.