Let $G$ be a group, and let $H$ be a subgroup of $G$. Show that the following are equivalent.

(i) $a^{-1} b^{-1} a b \in H$ for all $a, b \in G$.

(ii) $H$ is a normal subgroup of $G$ and $G / H$ is abelian.

Hence find all abelian quotient groups of the dihedral group $D_{10}$ of order 10 .