# generalized dihedral group

Let $A$ be an abelian group^{}.
The *generalized dihedral group* $\mathrm{Dih}(A)$
is the semidirect product^{} $A\u22ca{C}_{2}$,
where ${C}_{2}$ is the cyclic group^{} of order $2$,
and the generator^{} (http://planetmath.org/Generator) of ${C}_{2}$ maps elements of $A$ to their inverses^{}.

If $A$ is cyclic, then $\mathrm{Dih}(A)$ is called a dihedral group^{}.
The finite dihedral group $\mathrm{Dih}({C}_{n})$ is commonly denoted by ${D}_{n}$ or ${D}_{2n}$
(the differing conventions being a source of confusion).
The infinite dihedral group $\mathrm{Dih}({C}_{\mathrm{\infty}})$ is denoted by ${D}_{\mathrm{\infty}}$,
and is isomorphic^{} to
the free product^{} ${C}_{2}*{C}_{2}$ of two cyclic groups of order $2$.

If $A$ is an elementary abelian $2$-group, then so is $\mathrm{Dih}(A)$.
If $A$ is not an elementary abelian $2$-group, then $\mathrm{Dih}(A)$ is non-abelian^{}.

The subgroup^{} $A\times \{1\}$ of $\mathrm{Dih}(A)$ is of index $2$,
and every element of $\mathrm{Dih}(A)$ that is not in this subgroup has order $2$.
This property in fact characterizes generalized dihedral groups,
in the sense that if a group $G$ has a subgroup $N$ of index $2$ such that all elements of the complement $G\setminus N$ are of order $2$,
then $N$ is abelian and $G\cong \mathrm{Dih}(N)$.

Title | generalized dihedral group |
---|---|

Canonical name | GeneralizedDihedralGroup |

Date of creation | 2013-03-22 14:53:28 |

Last modified on | 2013-03-22 14:53:28 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E22 |

Synonym | generalised dihedral group |

Related topic | DihedralGroup |

Defines | infinite dihedral group |

Defines | infinite dihedral |