# Number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size

From Groupprops

## Statement

Suppose is a natural number. Then, there exists a polynomial function of degree such that, for any prime power , the number of conjugacy classes in the general affine group (i.e., the general affine group of degree over the finite field of size ) is .

## General observations

Below, we list some general observations about the polynomial in giving number of conjugacy classes in .

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

Degree of polynomial | |

Leading coefficient of polynomial | 1, i.e., it is always a monic polynomial |

Factors of polynomial | no common factors to all polynomials |

Coefficients of polynomial | The polynomial is an integer-valued polynomial, i.e., it sends integers to integers. Is it also true that all coefficients of the polynomial are integers? Seems so from first few examples |

## Particular cases

(degree of general affine group, degree of polynomial) | polynomial of giving number of conjugacy classes in | More information |
---|---|---|

1 | element structure of general affine group of degree one over a finite field | |

2 | element structure of general affine group of degree two over a finite field | |

3 | element structure of general affine group of degree three over a finite field | |

4 | element structure of general affine group of degree four over a finite field | |

5 | ? | element structure of general affine group of degree five over a finite field |

6 | ? | element structure of general affine group of degree six over a finite field |

7 | ? |