Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$

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

The forward direction is quite easy. Suppose $H$ is a subgroup. Then by closure, $ab \in H$ for any $a,b \in H$. Every element has an inverse. Hence, if $b \in H $, then $b^{-1} \in H$. Hence, by closure again, $ab^{-1} \in H$.

Backward direction, suppose $ab^{-1} \in H $ for any $a,b \in H$. Let $a=b$. Then we have $bb^{-1}=1_H\in H$. Let $a=1_H$ , we have $1_Hb^{-1}=b^{-1} \in H$. I don't know how to show closure.

Reference: Fraleigh p. 58 Question 5.45 in A First Course in Abstract Algebra

Solution 1:

Take $a$, $b$ in $H$ then $c=b^{-1}\in H$, so that $a c^{-1}=a (b^{-1})^{-1}=ab\in H$.