How to prove that if group $G$ is abelian, $H = \{x \in G : x = x^{-1}\}$ is a subgroup? $G$ is abelian, $H = \{x \in G : x = x^{-1}\}$ is a subgroup?
I know to prove that a subset $H$ of group $G$ be a subgroup, one needs to (i) prove $\forall x,y \in H:x \circ y \in H$ and (ii) $\forall x \in H:x^{-1} \in H.$

Consider the map $\varphi:G\to G$ given by $\varphi(x)=x^2$. Since $G$ is abelian, then $\varphi$ is a homomorphism, and it's readily seen that $H$ is precisely the kernel of $\varphi$. Thus, $H$ is a subgroup of $G$.
