Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statements are equivalent.\\
(A) For every nontrivial subgroup $K$ of $G$, the subgroup $K \cap H$ is also nontrivial.\\
(B) $H$ contains every nontrivial minimal subgroup of $G$ and every element of the quotient group $G / H$ has finite order.