# examples of semidirect products of groups

Suppose $H=\mathbb{Z}/n\mathbb{Z}$ and let $r$ be a generator for $H$. Let $Q=\mathbb{Z}/2\mathbb{Z}=$. Define $\theta:Q\to\operatorname{Aut}(H)$ by $\theta(s)(r)=r^{-1}$. Let $G=H\rtimes_{\theta}Q$. Then in $G$,

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

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

 $\displaystyle r^{n}=s^{2}=1$ $\displaystyle srs=r^{-1}$

and thus $G=\mathcal{D}_{2n}$, 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 $G\rtimes\operatorname{Aut}(G)$ under the identity map from $\operatorname{Aut}(G)$ to itself.

Title examples of semidirect products of groups ExamplesOfSemidirectProductsOfGroups 2013-03-22 17:22:52 2013-03-22 17:22:52 rm50 (10146) rm50 (10146) 4 rm50 (10146) Example msc 20E22