# Difference between revisions of "Element structure of dihedral group:D8"

From Groupprops

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group = dihedral group:D8| | group = dihedral group:D8| | ||

connective = of}} | connective = of}} | ||

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+ | We denote the identity element by <math>e</math>. | ||

{{#lst:dihedral group:D8|multiplication table}} | {{#lst:dihedral group:D8|multiplication table}} |

## Revision as of 01:23, 9 August 2010

This article gives specific information, namely, element structure, about a particular group, namely: dihedral group:D8.

View element structure of particular groups | View other specific information about dihedral group:D8

We denote the identity element by .

## Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Conjugacy class | Size of conjugacy class | Order of elements in conjugacy class | Centralizer of first element of class |
---|---|---|---|

1 | 1 | whole group | |

1 | 2 | whole group | |

2 | 2 | -- one of the Klein four-subgroups of dihedral group:D8 | |

2 | 2 | -- one of the Klein four-subgroups of dihedral group:D8 | |

2 | 4 | -- the cyclic maximal subgroup of dihedral group:D8 |

The equivalence classes up to automorphisms are:

Equivalence class under automorphisms | Size of equivalence class | Number of conjugacy classes in it | Size of each conjugacy class |
---|---|---|---|

1 | 1 | 1 | |

1 | 1 | 1 | |

4 | 2 | 2 | |

2 | 1 | 2 |

### Convolution algebra on conjugacy classes

## Order statistics

Number | Elements of order exactly that number | Number of such elements | Number of conjugacy classes of such elements | Number of elements whose order divides that number | Number of conjugacy classes whose element order divides that number |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | |

2 | 5 | 3 | 6 | 4 | |

4 | 2 | 1 | 8 | 5 |