If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers being isomorphic to the even integers, so that reasoning doesn't work. However in this case the even integers are not a quotient of the integers.
edit: I realize now that $G/N$ is not necessarily isomorphic to a proper subgroup of $G$, just a subgroup of $G$.

Take any infinite dimensional vector space $V$, and let $W$ be any subspace of $V$ whose direct complement has the same dimension as $V$. For example, any finite dimensional subspace.
Then $V/W$ must have the same dimension as $V$.
(And now note that vector spaces are Abelian groups, and that subspaces are subgroups...)
