# 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 $G=H{\u22ca}_{\theta}Q$. 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 $G={\mathcal{D}}_{2n}$, 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 $G\u22ca\mathrm{Aut}(G)$ 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 |