Non-Abelian groups and subgroups Need to find example for non-abelian group $G$ for which $A \subset G,\;A=\{g\in G \mid g^{-1}=g\}$ is not a subgroup of $G$.
Can you please help me find such a $G$?

A symmetric group is an easy answer.  Take $S_3$, then the permutations $(1 \ 2)$ and $(2 \ 3)$ are each their own inverse (I'm using cycle notation, btw), but their product $(1 \ 2)(2 \ 3) = (1 \ 2 \ 3)$ is not.
