# Infinite dihedral group

From Groupprops

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## Definition

The **infinite dihedral group**, denoted , is defined by the following presentation:

.

Here denotes the identity element.

Equivalently, it is the generalized dihedral group corresponding to the additive group of integers.

## Related groups

- Generalized dihedral group for additive group of 2-adic integers
- Generalized dihedral group for 2-quasicyclic group

## Arithmetic functions

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

order ((number of elements, equivalently, cardinality or size of underlying set) | Infinite (countable) | Not a finite group. |

exponent | Infinite | Not a periodic group. |

derived length | 2 | The group of integers is a subgroup of index two (explicitly, the infinite dihedral group is a semidirect product of (isomorphic to the [group of integers]])and (a group of order two). |

nilpotency class | -- | -- |

Fitting length | 2 | The Fitting subgroup is the group of integers . |

Frattini length | 1 | The Frattini subgroup is trivial, because the maximal subgroups include and subgroups of the form for a prime number. The intersection of all these is trivial. |

subgroup rank of a group | 2 | The whole group is 2-generated, so the subgroup rank is at least 2. Any subgroup is either inside or has a subgroup of index two inside . Therefore, every subgroup is either cyclic or dihedral, and thus every subgroup is 2-generated. |

## Group properties

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

abelian group | No | |

centerless group | Yes | |

nilpotent group | No | It is a nontrivial centerless group. |

hypercentral group | No | |

group satisfying normalizer condition | No | The subgroup generated by is proper and self-normalizing. |

residually nilpotent group | Yes | |

hypocentral group | Yes | follows from being residually nilpotent |

metacyclic group | Yes | |

supersolvable group | Yes | |

polycyclic group | Yes | |

metabelian group | Yes | |

solvable group | Yes | |

finite group | No | |

2-generated group | Yes | |

finitely generated group | Yes | |

residually finite group | Yes | |

Hopfian group | Yes | finitely generated and residually finite implies Hopfian |

group with finitely many homomorphisms to any finite group | Yes | finitely generated implies finitely many homomorphisms to any finite group |

group in which every subgroup of finite index has finitely many automorphic subgroups | Yes | finitely generated implies every subgroup of finite index has finitely many automorphic subgroups |