# Element structure of dihedral groups

This article gives specific information, namely, element structure, about a family of groups, namely: dihedral group.

View element structure of group families | View other specific information about dihedral group

This article discusses the element structure of the dihedral group of degree and order , given by the presentation:

Here denotes the identity element.

## Summary

The information below is for .

Item | Value |
---|---|

order | |

conjugacy class sizes | Case odd: 1 (1 time), 2 ( times), (1 time) Case even: 1 (2 times), 2 ( times), (2 times) |

number of conjugacy classes | if odd, if even |

number of orbits under automorphism group | where is the divisor count function if , 2 if |

order statistics | Case odd: of order for , of order 2 |

## Particular cases

(degree) | (order) | Degree parity | Group | Second part of GAP ID | number of conjugacy classes ( if odd, if even) | number of orbits under automorphism group ( if , 2 if ) | element structure page |
---|---|---|---|---|---|---|---|

1 | 2 | odd | cyclic group:Z2 | 1 | 2 | 2 | element structure of cyclic group:Z2 |

2 | 4 | even | Klein four-group | 2 | 4 | 2 | element structure of Klein four-group |

3 | 6 | odd | symmetric group:S3 | 1 | 3 | 3 | element structure of symmetric group:S3 |

4 | 8 | even | dihedral group:D8 | 3 | 5 | 4 | element structure of dihedral group:D8 |

5 | 10 | odd | dihedral group:D10 | 1 | 4 | 3 | element structure of dihedral group:D10 |

6 | 12 | even | dihedral group:D12 | 4 | 6 | 5 | element structure of dihedral group:D12 |

7 | 14 | odd | dihedral group:D14 | 1 | 5 | 3 | element structure of dihedral group:D14 |

8 | 16 | even | dihedral group:D16 | 7 | 7 | 5 | element structure of dihedral group:D16 |

9 | 18 | odd | dihedral group:D18 | 1 | 6 | 4 | element structure of dihedral group:D18 |

10 | 20 | even | dihedral group:D20 | 4 | 8 | 5 | element structure of dihedral group:D20 |

## Odd degree case

This is the case is odd, so is twice an odd number.

### Conjugacy class structure

Nature of conjugacy class | Size of each conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|

Identity element | 1 | 1 | 1 |

Non-identity elements in cyclic subgroup , where each element and its inverse form a conjugacy class | 2 | ||

Elements outside the cyclic subgroup , all form a single conjugacy class | 1 | ||

Total | -- | (number of conjugacy classes) | (order of group) |

### Order information

We have the following number of elements of various orders:

Order type | Number of elements of that order | Specifics about the elements | Number of such order types | Total number of elements |
---|---|---|---|---|

a divisor of (hence is odd)</math> | , is the Euler totient function | These are all the generators of the cyclic subgroup | where is the divisor count function. | which becomes |

2 | All the elements outside | 1 |

### Equivalence classes up to automorphism

It turns out that the equivalence classes up to automorphism are given precisely by order of elements, i.e., two elements are automorphic elements if and only if they have the same order. Equivalently, two elements are automorphic if and only if they generate the same cyclic subgroup. More explicitly:

Description of equivalence class | Number of conjugacy classes per equivalence class | Size of each conjugacy class | Total number of elements in each equivalence class | Number of equivalence classes | Total number of conjugacy classes | Total number of elements across equivalence classes |
---|---|---|---|---|---|---|

Identity element | 1 | 1 | 1 | 1 | 1 | 1 |

For each nontrivial divisor of , the elements of order (generators of ) form one equivalence class | 2 | , is the Euler totient function | , is the divisor count function | |||

Elements outside the cyclic subgroup , of order 2 | 1 | 1 | 1 | |||

Total | -- | -- | -- |

## Even degree case

Suppose , and is the dihedral group of order . Then, has the following conjugacy classes (a total of conjugacy classes):

### Conjugacy class structure

Nature of conjugacy class | Size of each conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|

Identity element | 1 | 1 | 1 |

Non-identity element of order 2 in | 1 | 1 | 1 |

Non-identity elements in cyclic subgroup , where each element and its inverse form a conjugacy class | 2 | ||

Elements outside , form two conjugacy classes, one for elements of the form , and the other for elements of the form | 2 | ||

Total | -- | (number of conjugacy classes) | (order of group) |

### Order information

We have the following number of elements of various orders:

Order | Number of elements of that order | Specifics about the elements |
---|---|---|

a divisor of other than 2 | , is the Euler totient function | These are all the generators of the cyclic subgroup |

2 | All the elements outside , as well as the element |

### Equivalence classes up to automorphism

Note that the data in the table below is not correct for the case , in which case we get the Klein four-group.

The equivalence classes up to automorphism are as follows:

Description of equivalence class | Number of conjugacy classes per equivalence class | Size of each conjugacy class | Total number of elements in each equivalence class | Number of equivalence classes | Total number of conjugacy classes | Total number of elements across equivalence classes |
---|---|---|---|---|---|---|

Identity element | 1 | 1 | 1 | 1 | 1 | 1 |

Element | 1 | 1 | 1 | 1 | 1 | 1 |

For each nontrivial divisor of other than 2, the elements of order (generators of ) form one equivalence class | 2 | , is the Euler totient function | , is the divisor count function | |||

Elements outside the cyclic subgroup , of order 2 | 2 | 1 | 2 | |||

Total | -- | -- | -- |