How can you tell if a normal subgroup induces a semidirect product?

Question

Suppose I have some (finite) group $G$ and a normal subgroup $N$. I know there's no full characterization of whether $G \cong N \rtimes G/N$, but are there well-known tests I can use to answer the question in common cases? Moreover, if $G$ is a semidirect product, then as we know $G \ge H \cong G/N$...but how do I explicitly find the elements of $H$? They lie in the cosets of $N$, but I don't know any good way to figure out, short of exhaustive testing, which the "right" elements are. Any suggestions for how to approach this appreciated.

Answer 1

A group $G$ and a normal subgroup $N$ determines a short exact sequence

$$1 \to N \to G \to G/N \to 1$$

and this short exact sequence exhibits $G$ as a semidirect product if and only if the map $G \to G/N$ splits, or equivalently if it has a right inverse. Given such a right inverse, the elements of $H$ are given by the image of the right inverse.

A simple example where this doesn't occur is the short exact sequence

$$1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1.$$

A sufficient condition for splitting is given by the Schur-Zassenhaus theorem: such a splitting always exists if $\gcd(|N|, |G/N|) = 1$. A simpler special case of this, using the Sylow theorems, is that such a splitting always exists if the order of $G/N$ is the order of a Sylow subgroup of $G$.