When is a group isomorphic to the product of normal subgroup and quotient group?

Question

Let $H$ be a normal subgroup in $G$. When is $G$ isomorphic to $H\times (G/H)$?

I think it's always true in the abelian case. Are there other rules?

Answer 1

It's not true in the abelian case. The smallest counterexample is $G = \mathbb{Z}_4, H = \mathbb{Z}_2$; the groups $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ are not isomorphic.

In general you need $H$ to be normal for this question makes sense. Then $G$ is an extension of $G/H$ by $H$. The classification of these is difficult in general and usually there are interesting extensions other than the trivial extension $H \times G/H$. When all three groups are abelian the classification is in terms of the Ext group $\text{Ext}^1(G/H, H)$.