Let $G$ be any group such that

$$G\cong G/H$$
where $H$ is a normal subgroup of $G$.

If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In what kind of group could I search for a counterexample ?

Look at $G=\bigoplus\limits_{i=1}^\infty\ \mathbb Z$ and the subgroup $H=\mathbb Z\ \oplus\ \bigoplus\limits_{i=2}^\infty\ 0$.