I feel very silly for asking this question:
Let $G$ be an abelian group. Show that $H=\{a\in G:a*a=a\}$ is a subgroup of $G$.
I did not get this question right, but this is what I managed to observe:
Notice that$a*a=a,$$a^{-1}*a*a=a^{-1}*a,$$e*a=e,$$a=e,$where $a^{-1}$ is the inverse of $a$, and $e$ is the identity element. Therefore, it must be the case that $H=\{e\}$, which is a subgroup of $G$.
The professor marked that conclusion wrong, however, and corrected it by commenting that this only showed that $e\in H$.
I am a little confused: look at how $H$ is defined. I believe this is just not an intuitive exercise; I never used the fact that $H$ is abelian, so I clearly must be missing something.