Semidirect Products and Extensions

Semidirect products generalize direct products by allowing one factor to act on the other, providing the main tool for constructing non-abelian groups from smaller ones.

Definition

Definition10.5Semidirect product

Let $N$ and $H$ be groups and $\varphi: H \to \mathrm{Aut}(N)$ a homomorphism. The semidirect product $N \rtimes_\varphi H$ is the set $N \times H$ with multiplication $(n_1, h_1)(n_2, h_2) = (n_1 \varphi(h_1)(n_2), h_1 h_2)$ .

This is a group with identity $(e_N, e_H)$ and inverse $(n,h)^{-1} = (\varphi(h^{-1})(n^{-1}), h^{-1})$ . The subgroup $N \times \{e\}$ is normal, $\{e\} \times H$ is a subgroup, and $G = N \rtimes H$ satisfies $G/N \cong H$ .

ExampleSemidirect product examples

$D_n = \mathbb{Z}/n\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$ where $\mathbb{Z}/2$ acts by inversion: $\varphi(1)(k) = -k$ . This is the dihedral group.
$S_n = A_n \rtimes \mathbb{Z}/2\mathbb{Z}$ for $n \geq 2$ (the sign homomorphism splits).
$\mathrm{Aff}(\mathbb{F}_p) = \mathbb{F}_p \rtimes \mathbb{F}_p^\times$ is the group of affine transformations $x \mapsto ax + b$ .
$\mathbb{Z}/3 \rtimes \mathbb{Z}/4$ (with the nontrivial action of order 2): a non-abelian group of order 12 distinct from $A_4$ and $D_6$ .

Group Extensions

Definition10.6Group extension

An extension of a group $N$ by a group $Q$ is a short exact sequence $1 \to N \to G \to Q \to 1$ . The extension splits if there exists a section $s: Q \to G$ with $\pi \circ s = \mathrm{id}_Q$ , in which case $G \cong N \rtimes Q$ .

Theorem10.4Schur-Zassenhaus theorem

If $N \trianglelefteq G$ is a normal subgroup with $\gcd(|N|, [G:N]) = 1$ (coprime orders), then the extension splits: $G \cong N \rtimes G/N$ . Moreover, all complements of $N$ in $G$ are conjugate.

RemarkNon-split extensions and group cohomology

Not all extensions split. The quaternion group $Q_8$ is an extension $1 \to \mathbb{Z}/2 \to Q_8 \to \mathbb{Z}/2 \times \mathbb{Z}/2 \to 1$ that does not split ( $Q_8$ has no subgroup isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}/2$ ). Non-split extensions are classified by the second cohomology group $H^2(Q, N)$ , which measures the "obstruction to splitting."