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

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| <math>A</matH> is the identity, <math>v \ne 0</math> || <math>\{ 1,1 \}</math> || <math>(x - 1)^2</math> || <math>x - 1</math> || <math>q^2 - 1</math> || 1 || <matH>q^2 - 1</math> || Yes || Yes | | <math>A</matH> is the identity, <math>v \ne 0</math> || <math>\{ 1,1 \}</math> || <math>(x - 1)^2</math> || <math>x - 1</math> || <math>q^2 - 1</math> || 1 || <matH>q^2 - 1</math> || Yes || Yes | ||

|- | |- | ||

− | | <math>A</math> is diagonalizable over <math>\mathbb{F}_q</math> with equal diagonal entries not equal to 1, hence a scalar. The value of <matH>v</math> does not affect the conjugacy class. || <math>\{a,a \}</math> where <math>a \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>(x - a)^2</math> where <math>a \in \mathbb{F}_q^\ast</math> || <math>x - a</math> where <math>a \in \mathbb{F}_q^\ast</math> || <math>q^2</math> || <math>q - 2</math> || <math>q^2(q - 2)</math> || Yes || Yes | + | | <math>A</math> is diagonalizable over <math>\mathbb{F}_q</math> with equal diagonal entries not equal to 1, hence a scalar. The value of <matH>v</math> does not affect the conjugacy class. || <math>\{a,a \}</math> where <math>a \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>(x - a)^2</math> where <math>a \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>x - a</math> where <math>a \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>q^2</math> || <math>q - 2</math> || <math>q^2(q - 2)</math> || Yes || Yes |

|- | |- | ||

| <math>A</math> is diagonalizable over <math>\mathbb{F}_{q^2}</math>, not over <math>\mathbb{F}_q</math>. Must necessarily have no repeated eigenvalues. The value of <math>v</matH> does not affect the conjugacy class. || Pair of conjugate elements of <math>\mathbb{F}_{q^2}</math> || <math>x^2 - ax + b</math>, irreducible || Same as characteristic polynomial || <math>q^3(q - 1)</math> || <math>q(q - 1)/2 = (q^2 - q)/2</math> || <math>q^4(q-1)^2/2</math> || Yes || No | | <math>A</math> is diagonalizable over <math>\mathbb{F}_{q^2}</math>, not over <math>\mathbb{F}_q</math>. Must necessarily have no repeated eigenvalues. The value of <math>v</matH> does not affect the conjugacy class. || Pair of conjugate elements of <math>\mathbb{F}_{q^2}</math> || <math>x^2 - ax + b</math>, irreducible || Same as characteristic polynomial || <math>q^3(q - 1)</math> || <math>q(q - 1)/2 = (q^2 - q)/2</math> || <math>q^4(q-1)^2/2</math> || Yes || No |

## Revision as of 23:40, 22 February 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 | 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) |

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

5 | 5 | 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 |