1. ## Subgroup

Show that a nonempty subset $H$ of a group $G$ is a subgroup of $G$ if and only if $ab^{-1} \in H$ for all $a, b \in H$.

2. ## Re: Subgroup

Originally Posted by alexmahone
Show that a nonempty subset $H$ of a group $G$ is a subgroup of $G$ if and only if $ab^{-1} \in H$ for all $a, b \in H$.
Clearly this condition must hold if your set is a subgroup (as subgroups are closed under multiplication).

If this condition holds then note that you have the identity in your set (take $a=b$), and that if $g\in H$ then $g^{-1} \in H$ (take $a=1$, $b=g$).

So the only thing to prove is that if $g, h\in H$ then $gh\in H$. However, as $h^{-1}\in H$, take $a=g$ and $b=h^{-1}$...and your done!