# Generalized dihedral group

From Groupprops

## Contents

## Definition

Suppose is an abelian group. The **generalized dihedral group** corresponding to is the external semidirect product of with the cyclic group of order two, with the non-identity element acting as the inverse map on .

Viewing this external semidirect product as an internal semidirect product, is an abelian normal subgroup of index two.

A *presentation* for is:

.

Note that the dihedral groups are special cases of generalized dihedral groups where the abelian group in question is a cyclic group.

## Relation with other properties

### Stronger properties

### Weaker properties

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | Twice the order of | |

exponent | Least common multiple of and the exponent of | |

derived length | if is an elementary abelian -group, otherwise. | |

nilpotency class | Frattini length of if is a -group, not defined otherwise. | |

max-length | One more than the max-length of . | |

composition length | One more than the composition length of . | |

chief length | One more than the chief length of . | |

minimum size of generating set | One more than the minimum size of generating set of . | |

number of subgroups | number of subgroups of plus sum of indices of subgroups of . | |

number of conjugacy classes | where and where is the set of squares in . |

## Group properties

Property | Satisfied | Explanation |
---|---|---|

Abelian group | True only if is an elementary abelian -group | |

Nilpotent group | True only if is a -group | |

Solvable group | Yes | |

Metabelian group | Yes |

## Subgroups

`Further information: Subgroup structure of generalized dihedral groups`

There are two kinds of subgroups of the generalized dihedral group with the abelian subgroup :

- Subgroups of : All of these are normal subgroups of . The number of such subgroups equals the number of subgroups of .
- Subgroups of containing an element outside : Suppose is such a subgroup. Then is a subgroup of index two in , and is the union of and a coset where , i.e., . Thus, to specify , it suffices to specify and the coset . Conversely, given
*any*subgroup and*any*coset with , we obtain a subgroup . The reason this is a subgroup is because has order two and acts on by the inverse map. The upshot is that, given , the number of possibilities for is the number of cosets of outside , which equals the number of cosets inside , which equals the index . Thus, the*total*number of possibilities for is the sum of subgroup indices over all subgroups of .