Element structure of special linear group:SL(2,3): Difference between revisions

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! Nature of conjugacy class  !! Eigenvalue pairs of all conjugacy classes !! Characteristic polynomials of all conjugacy classes !! Minimal polynomials of all conjugacy classes !! Size of conjugacy class!! Number of such conjugacy classes !! Total number of elements !! Semisimple? !! Diagonalizable over <math>\mathbb{F}_q</math>? !! Splits in <math>SL_2</math> relative to <math>GL_2</math>? !! Representative matrices (one per conjugacy class)
! Nature of conjugacy class  !! Eigenvalue pairs of all conjugacy classes !! Characteristic polynomials of all conjugacy classes !! Minimal polynomials of all conjugacy classes !! Size of conjugacy class!! Number of such conjugacy classes !! Total number of elements !! Semisimple? !! Diagonalizable over <math>\mathbb{F}_q</math>? !! Splits in <math>SL_2</math> relative to <math>GL_2</math>? !! Representative matrices (one per conjugacy class)
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| Diagonalizable over [[field:F3]] with ''distinct'' diagonal entries || -- || -- || -- || NA || 0 || 0 || NA || NA || NA || --
| Scalar || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 - x + 1</math> || <math>x - 1</math> or <math>x + 1</math> || 1 || 2 || 2 || Yes || Yes || No || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}</math>
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| Not diagonal, Jordan block of size two || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 - x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 - x + 1</math>  || 4 || 4 || 16 || No || No || Yes || <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math>
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| Diagonalizable over [[field:F9]], not over [[field:F3]]. Must necessarily have no repeated eigenvalues. || pair of square roots of <math>-1</math> in [[field:F9]] || <math>x^2 + 1</math> || <math>x^2 + 1</math> || 6 || 1 || 6 || Yes ||  No || No || <math>\begin{pmatrix}0 & -1\\ 1 & 0\\\end{pmatrix}</math>
| Diagonalizable over [[field:F9]], not over [[field:F3]]. Must necessarily have no repeated eigenvalues. || pair of square roots of <math>-1</math> in [[field:F9]] || <math>x^2 + 1</math> || <math>x^2 + 1</math> || 6 || 1 || 6 || Yes ||  No || No || <math>\begin{pmatrix}0 & -1\\ 1 & 0\\\end{pmatrix}</math>
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| Scalar || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 - x + 1</math> || <math>x - 1</math> or <math>x + 1</math> || 1 || 2 || 2 || Yes || Yes || No || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}</math>
| Diagonalizable over [[field:F3]] with ''distinct'' diagonal entries || -- || -- || -- || NA || 0 || 0 || NA || NA || NA || --
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| Not diagonal, Jordan block of size two || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 - x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 - x + 1</math>  || 4 || 4 || 16 || No || No || Yes || <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math>
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| Total || NA || NA || NA || NA || 7 || 24 || 8 || 2 || 16 || NA
| Total || NA || NA || NA || NA || 7 || 24 || 8 || 2 || 16 || NA
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|}

Revision as of 00:37, 30 October 2010

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,3).
View element structure of particular groups | View other specific information about special linear group:SL(2,3)

This article gives detailed information about the element structure of special linear group:SL(2,3).

See also element structure of special linear group of degree two.

Conjugacy class structure

Conjugacy classes

Conjugacy class representative Conjugacy class size List of all elements of conjugacy class
(1001) 1 (1001)
(−100−1) 1 (−100−1)
(0−110) 6 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
(1101) 4 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
(1−101) 4 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
(1−101) 4 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
(−1−10−1) 4 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Relationship with conjugacy class structure for an arbitrary special linear group of degree two

Further information: element structure of special linear group of degree two

Nature of conjugacy class Eigenvalue pairs of all conjugacy classes Characteristic polynomials of all conjugacy classes Minimal polynomials of all conjugacy classes Size of conjugacy class Number of such conjugacy classes Total number of elements Semisimple? Diagonalizable over Fq? Splits in SL2 relative to GL2? Representative matrices (one per conjugacy class)
Scalar {1,1} or {−1,−1} x2−2x+1 or x2−x+1 x−1 or x+1 1 2 2 Yes Yes No (1001) and (−100−1)
Not diagonal, Jordan block of size two {1,1} or {−1,−1} x2−2x+1 or x2−x+1 x2−2x+1 or x2−x+1 4 4 16 No No Yes (1101), (1−101), (−110−1), (−1−10−1)
Diagonalizable over field:F9, not over field:F3. Must necessarily have no repeated eigenvalues. pair of square roots of −1 in field:F9 x2+1 x2+1 6 1 6 Yes No No (0−110)
Diagonalizable over field:F3 with distinct diagonal entries -- -- -- NA 0 0 NA NA NA --
Total NA NA NA NA 7 24 8 2 16 NA
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