Difference between revisions of "Element structure of general affine group of degree two over a finite field"
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| <math>A</math> diagonalizable over <math>\mathbb{F}_q</math> with ''distinct'' diagonal entries, one of which is 1, <math>v</math> is not in the image of <math>A - 1</matH> || <math>1,\mu</math>, <math>\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>x^2 - (\mu + 1)x + \mu</math> || Same as characteristic polynomial || <math>q(q + 1)(q^2 - q)</math> || <math>q - 2</math> || <math>q^2(q+1)(q - 1)(q-2)</math> || Yes || Yes | | <math>A</math> diagonalizable over <math>\mathbb{F}_q</math> with ''distinct'' diagonal entries, one of which is 1, <math>v</math> is not in the image of <math>A - 1</matH> || <math>1,\mu</math>, <math>\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>x^2 - (\mu + 1)x + \mu</math> || Same as characteristic polynomial || <math>q(q + 1)(q^2 - q)</math> || <math>q - 2</math> || <math>q^2(q+1)(q - 1)(q-2)</math> || Yes || Yes | ||
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− | | <math>A</math> diagonalizable over <math>\mathbb{F}_q</math> with ''distinct'' diagonal entries, neither of which is 1 || <math>\lambda, \mu</math> (interchangeable) distinct elements of <math>\mathbb{F}_q^\ast</math> || <math>x^2 - (\lambda + \mu)x + \lambda \mu</math> || Same as characteristic polynomial || <math>q^3(q+1)</math>|| <math>(q - 2)(q - 3)/2 </math> || <math>q^3(q+1)(q-2)(q-3)/2</math> || Yes || Yes | + | | <math>A</math> diagonalizable over <math>\mathbb{F}_q</math> with ''distinct'' diagonal entries, neither of which is 1 || <math>\lambda, \mu</math> (interchangeable) distinct elements of <math>\mathbb{F}_q^\ast</math>, neither equal to 1 || <math>x^2 - (\lambda + \mu)x + \lambda \mu</math> || Same as characteristic polynomial || <math>q^3(q+1)</math>|| <math>(q - 2)(q - 3)/2 </math> || <math>q^3(q+1)(q-2)(q-3)/2</math> || Yes || Yes |
|- | |- | ||
! Total !! NA !! NA !! NA !! NA !! <math>q^2 + q - 1</math> !! <math>q^2(q^2 - 1)(q^2 - q)</math> !! !! | ! Total !! NA !! NA !! NA !! NA !! <math>q^2 + q - 1</math> !! <math>q^2(q^2 - 1)(q^2 - q)</math> !! !! | ||
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<section end="conjugacy class structure"/> | <section end="conjugacy class structure"/> |
Revision as of 23:36, 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
![]() |
![]() |
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 ![]() |
Is ![]() ![]() |
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1 | 1 | 1 | Yes | Yes |
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1 | ![]() |
Yes | Yes |
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Yes | Yes |
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Pair of conjugate elements of ![]() |
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Same as characteristic polynomial | ![]() |
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Yes | No |
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Same as characteristic polynomial | ![]() |
1 | ![]() |
No | No |
![]() ![]() ![]() |
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Same as characteristic polynomial | ![]() |
1 | ![]() |
No | No |
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Same as characteristic polynomial | ![]() |
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No | No |
![]() ![]() ![]() ![]() |
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Same as characteristic polynomial | ![]() |
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Yes | Yes |
![]() ![]() ![]() ![]() |
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Same as characteristic polynomial | ![]() |
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Yes | Yes |
![]() ![]() |
![]() ![]() |
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Same as characteristic polynomial | ![]() |
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Yes | Yes |
Total | NA | NA | NA | NA | ![]() |
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