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

From Groupprops

(→Convolution algebra on conjugacy classes) |
|||

Line 3: | Line 3: | ||

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

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

− | + | {{fblike}} | |

− | We denote the identity element by <math>e</math>. | + | We denote the identity element by <math>e</math>. Below is the multiplication table, with the row element multiplied on the left and the column element on the right. |

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

## Revision as of 01:26, 22 June 2011

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 . Below is the multiplication table, with the row element multiplied on the left and the column element on the right.

## Summary

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

order of the whole group (total number of elements) | 8 |

conjugacy class sizes | 1,1,2,2,2 maximum: 2, number of conjugacy classes: 5, lcm: 2 |

order statistics | 1 of order 1, 5 of order 2, 2 of order 4 maximum: 4, lcm (exponent of the whole group): 4 |

## 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 and power information

### Directed power graph

Below is a trimmed version of the directed power graph of the group. There is a dark edge from one vertex to another if the latter is the square of the former. A dashed edge means that the latter is an odd power of the former. We remove all the loops.

### 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 |

### Power statistics

Number | powers that are not powers for any larger divisor of the group order | Number of such elements | Number of conjugacy classes of such elements | Number of powers | Number of conjugacy classes of powers |
---|---|---|---|---|---|

1 | 6 | 3 | 8 | 5 | |

2 | 1 | 1 | 2 | 2 | |

4 | -- | 0 | 0 | 1 | 1 |

8 | 1 | 1 | 1 | 1 |