# Element structure of general affine group of degree one over a finite field

From Groupprops

## Contents |

This article gives specific information, namely, element structure, about a family of groups, namely: general affine group of degree one.

View element structure of group families | View other specific information about general affine group of degree one

This article describes the element structure of the general affine group of degree one over a finite field. We denote the field size by the letter and the characteristic of the field by the letter . Note that is a prime power with underlying prime .

## Summary

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

number of conjugacy classes | equals number of irreducible representations, see also linear representation theory of general affine group of degree one over a finite field |

order | |

exponent | |

conjugacy class size statistics | size 1 (1 class), size (1 class), size ( classes) |

## Conjugacy class structure

All the elements of this group are of the form:

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

1 | 1 | 1 | |

(conjugacy class is independent of choice of ) | 1 | ||

(conjugacy class is determined completely by choice of and is independent of choice of ; in other words, each conjugacy class is a coset of the subgroup of translations) | |||

Total (--) | -- |