A questions on the groups by a copy of $\Bbb Z$ Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $G/H$ contains a copy of $\Bbb Z$. Is this true that $G$ contains a copy of $\Bbb Z$? ($\Bbb Z$ is the group of integer numbers)

Another viewpoint using a bit simpler terms.  Containing a copy of $\mathbb Z$ means that it has an element of infinite order,   so we have $o(g+H)=\infty$ for some $g\in G$.   Now,  under the natural projection map, $o(\phi (g))|o(g)$so $\infty |o(g)$,  hence $o(g)=\infty$
