Direct product and quotient groups

Question

I am trying to convince myself that the following statement is true:

Let $G, F$ be two group and $H$ be a normal subgroup of $G$. Then $$G\cong H\oplus F\quad\text{if and only if}\quad G/H\cong F$$ Where $\oplus$ symbolizes the direct product $H\oplus F=\{(h,f):h\in H,f\in F\}$.

The case $(\Rightarrow)$ is trivial since we only have to apply the First Isomorphism Theorem to the natural projection map $H\oplus F\to F$.

Unfortunately I am not able to show that the converse also holds.

Thanks in advance

Answer 1

The converse is false. Let $G=\mathbb{Z}_4$ and $H = \{0,2\}$. Then $G/H \cong \mathbb{Z}_2$ but of course $G$ cannot be isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$.