Let $G$ be a group and let $\Phi\colon G \to G$ be an isomorphism. Define $H = \{ a \in G\ |\ \Phi(a) = a^{-1} \}$

Question

It asks to prove that if $H$ is a subgroup of $G$, then $G$ is abelian.

Solution: I showed that for every $a$ and $b$ in $H$, $a$ and $b$ commute. But how do I generalize to elements in $G$ NOT in $H$?