Prove that, A non- empty subset $\mathrm{H}$ of a group $(\mathrm{G}, *)$ to be a subgroup is $\mathrm{a} \in \mathrm{H}, \mathrm{b} \in \mathrm{H} \rightarrow \mathrm{a}^* \mathrm{~b}^{-1} \in \mathrm{H}$

Solution:

Let H be a subgroup of the group G and $b \in H, a \in H$ then $b \in H$ ie. $b^{-1} \in H \cdot$ [Existence of inverse]

$\therefore \quad a \in H, b \in H, \quad b^{-1} \in H, a^{-1} \in H$

$\Rightarrow a b^{-1} \in H[B y$ …