# Difference between revisions of "Element structure of general affine group of degree two over a finite field"

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− | ! <math>q</math> (field size) !! <math>p</math> (underlying prime, field characteristic) !! general affine group <math>GL(2,q)</math> !! [[order]] of the group (= <math>q^2(q^2 - 1)(q^2 - q)</math>)!! number of conjugacy classes (= <math>q^2 + q - 1</math>) !! element structure page | + | ! <math>q</math><br> (field size) !! <math>p</math> <br>(underlying prime, field characteristic) !! <math>r = \log_pq</math> !! general affine group <math>GL(2,q)</math> !! [[order]] of the group (= <math>q^2(q^2 - 1)(q^2 - q)</math>)!! number of conjugacy classes (= <math>q^2 + q - 1</math>) !! element structure page |

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− | | 2 || 2 || [[symmetric group:S4]] || 24 || 5 || [[element structure of symmetric group:S4]] | + | | 2 || 2 || 1 || [[symmetric group:S4]] || 24 || 5 || [[element structure of symmetric group:S4]] |

|- | |- | ||

− | | 3 || 3 || [[general affine group:GA(2,3)]] || 432 || 11 || [[element structure of general affine group:GA(2,3)]] | + | | 3 || 3 || 1 || [[general affine group:GA(2,3)]] || 432 || 11 || [[element structure of general affine group:GA(2,3)]] |

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− | | 4 || 2 || [[general affine group:GA(2,4)]] || 2880 || 19 || | + | | 4 || 2 || 2 || [[general affine group:GA(2,4)]] || 2880 || 19 || |

|- | |- | ||

− | | 5 || 5 || [[general affine group:GA(2,5)]] || 12000 || 29 || | + | | 5 || 5 || 1 || [[general affine group:GA(2,5)]] || 12000 || 29 || |

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## Latest revision as of 22:50, 1 March 2012

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

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

This article gives the element structure of the general affine group of degree two over a finite field. Similar structure works over an infinite field or a field of infinite characteristic, with suitable modification. For more on that, see element structure of general affine group of degree two over a field.

The discussion here builds upon the discussion of element structure of general linear group of degree two over a finite field.

## Summary

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

order | |

exponent | ? |

number of conjugacy classes |

## Particular cases

(field size) |
(underlying prime, field characteristic) |
general affine group | order of the group (= ) | number of conjugacy classes (= ) | element structure page | |
---|---|---|---|---|---|---|

2 | 2 | 1 | symmetric group:S4 | 24 | 5 | element structure of symmetric group:S4 |

3 | 3 | 1 | general affine group:GA(2,3) | 432 | 11 | element structure of general affine group:GA(2,3) |

4 | 2 | 2 | general affine group:GA(2,4) | 2880 | 19 | |

5 | 5 | 1 | general affine group:GA(2,5) | 12000 | 29 |

## Conjugacy class structure

There is a total of elements, and there are conjugacy classes of elements.

The conjugacy class structure is closely related to that of -- see Element structure of general linear group of degree two over a finite field#Conjugacy class structure.

We describe a generic element of in the form:

where is the dilation component and is the translation component.

Consider the quotient mapping , which sends the generic element to . Under this mapping, the following is true:

- For those conjugacy classes of comprising elements that do not have 1 as an eigenvalue, the full inverse image of the conjugacy class is a single conjugacy class in . In other words, the translation component does not matter.
- For those conjugacy classes of comprising elements that do have 1 as an eigenvalue, the conjugacy class splits into two depending on whether is in the image of .

Nature of conjugacy class | Eigenvalues | Characteristic polynomial of | Minimal polynomial of | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Is semisimple? | Is diagonalizable over ? |
---|---|---|---|---|---|---|---|---|

is the identity, | 1 | 1 | 1 | Yes | Yes | |||

is the identity, | 1 | Yes | Yes | |||||

is diagonalizable over with equal diagonal entries not equal to 1, hence a scalar. The value of does not affect the conjugacy class. | where | where | where | Yes | Yes | |||

is diagonalizable over , not over . Must necessarily have no repeated eigenvalues. The value of does not affect the conjugacy class. | Pair of conjugate elements of | , irreducible | Same as characteristic polynomial | Yes | No | |||

has Jordan block of size two, with repeated eigenvalue equal to 1, is in the image of | Same as characteristic polynomial | 1 | No | No | ||||

has Jordan block of size two, with repeated eigenvalue equal to 1, is not in the image of | Same as characteristic polynomial | 1 | No | No | ||||

has Jordan block of size two, with repeated eigenvalue not equal to 1 | (multiplicity two) where | where | Same as characteristic polynomial | No | No | |||

diagonalizable over with distinct diagonal entries, one of which is 1, is in the image of |
, | Same as characteristic polynomial | Yes | Yes | ||||

diagonalizable over with distinct diagonal entries, one of which is 1, is not in the image of |
, | Same as characteristic polynomial | Yes | Yes | ||||

diagonalizable over with distinct diagonal entries, neither of which is 1 |
(interchangeable) distinct elements of , neither equal to 1 | Same as characteristic polynomial | Yes | Yes | ||||

Total | NA | NA | NA | NA |