Subgroup

**alexmahone** · July 7th 2011, 06:39 AM

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 $a, b \in H$ .

**Swlabr** · July 7th 2011, 07:00 AM

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!

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