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

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

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

This article describes the element structure of the general semilinear group of degree one. Recall that, for a field , the group is defined as:

where is the multiplicative group of , is the prime subfield of , and denotes the Galois group of over .

We are interested specifically in the case where is a finite field of size , in which case the group is written as . Suppose is a prime power with underlying prime , so that for a positive integer . is the characteristic of . In this case, is cyclic of order (see multiplicative group of a finite field is cyclic) and is cyclic of order (generated by the Frobenius map ).

Thus, is a metacyclic group of order with presentation:

(here denotes the identity element).

Note that if , the group is the same as and is cyclic of order .

## Summary

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

order | |

number of conjugacy classes | (see explicit polynomials for fixed values of below) |

### Number of conjugacy classes formulas

For every fixed value of , the number of conjugacy classes simply becomes a polynomial in . The values of these polynomials for small are listed below:

Polynomial in giving number of conjugacy classes | |
---|---|

1 | |

2 | |

3 |

## Particular cases

(field size) | (field characteristic) | general semilinear group | order of the group (= ) | number of conjugacy classes (polynomial in dependent on : for , for , for ) | element structure page | |
---|---|---|---|---|---|---|

2 | 2 | 1 | trivial group | 1 | 1 | -- |

3 | 3 | 1 | cyclic group:Z2 | 2 | 2 | element structure of cyclic group:Z2 |

4 | 2 | 2 | symmetric group:S3 | 6 | 3 | element structure of symmetric group:S3 |

5 | 5 | 1 | cyclic group:Z4 | 4 | 4 | element structure of cyclic group:Z4 |

7 | 7 | 1 | cyclic group:Z6 | 6 | 6 | element structure of cyclic group:Z6 |

8 | 2 | 3 | general semilinear group:GammaL(1,8) | 21 | 5 | element structure of general semilinear group:GammaL(1,8) |

9 | 3 | 2 | semidihedral group:SD16 | 16 | 7 | element structure of semidihedral group:SD16 |

## Conjugacy class structure

### For generic r

This table is complicated! We consider conjugacy classes *in* the multiplicative group and conjugacy classes *outside* the multiplicative group. Note that .

Nature of conjugacy class | Number of rows of this type | Size of conjugacy class for each row | Number of such conjugacy classes for each row | Total number of elements for each row | Total number of conjugacy classes | Total number of elements |
---|---|---|---|---|---|---|

in the multiplicative group and in the prime subfield | 1 | 1 | ||||

(one such row for every positive divisor of with ) generates precisely the subfield of size for dividing , | where is the divisor count function | |||||

(one such row for every positive divisor of with ) outside the multiplicative group, and induces an automorphism of raising to the power of where is relatively prime to | ||||||

Total | -- | -- | -- | -- | (equal total number of conjugacy classes) | (equals order of group) |

### For r = 2

We take :

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

in the multiplicative group and in the prime subfield | 1 | ||

outside the prime subfield | 2 | ||

outside the multiplicative group | |||

Total | -- | (equals number of conjugacy classes in the group) | (equals order of the whole group) |

### For r = 3

We take :

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

in the multiplicative group and in the prime subfield | 1 | ||

outside the prime subfield | 3 | ||

outside the multiplicative group | |||

Total | -- | (equals number of conjugacy classes in the group) | (equals order of the whole group) |