examples of semidirect products of groups

Suppose $H=\mathbb{Z}/n\mathbb{Z}$ and let $r$ be a generator for $H$. Let . Define $\theta :Q\to \mathrm{Aut}(H)$ by $\theta (s)(r)=r^{-1}$. Let $G=H{⋊}_{\theta }Q$. Then in $G$,

$$srs=sr{s}^{-1}=\theta (s)(r)=r^{-1}$$

by the canonical equivalence of inner and outer semidirect products. So $G$ has $2n$ elements, two generators $r,s$ satisfying

$$r^n=s^2=1$$

$$srs=r^{-1}$$

and thus $G={𝒟}_{2n}$, the $n^{\mathrm{th}}$ dihedral group.

If instead $H=\mathbb{Z}$, the result is the infinite dihedral group.

As another example, if $G$ is a group, then the holomorph of $G$ is $G⋊\mathrm{Aut}(G)$ under the identity map from $\mathrm{Aut}(G)$ to itself.

Title	examples of semidirect products of groups
Canonical name	ExamplesOfSemidirectProductsOfGroups
Date of creation	2013-03-22 17:22:52
Last modified on	2013-03-22 17:22:52
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	4
Author	rm50 (10146)
Entry type	Example
Classification	msc 20E22