Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $ G\cong N \times G/N. $

Question

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $$ G\cong N \times G/N. $$

I tried to prove this claim, but then it seems that since $G$ is abelian then every subgroup is normal. So every group can be factored as above. But I did not find a counter example.

Any suggestions?

Thanks for helpers!

Answer 1

Take the cyclic group $G=C_{p^2}$ and $N=C_p$ for $p$ prime. Then it is not true that $$ G=C_{p^2}\cong C_p\times C_p=N\times G/N, $$ because the group $C_p\times C_p$ is not cyclic.

Answer 2

Hint: Is $\mathbb{Z}_4$ isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$?

Answer 3

Take $G=\mathbb{Z}\times\mathbb{Z}$ and $N=0\times 2\mathbb{Z}$, $G/N$ has a non-trivial element of finite order, but not $G$.