Difference between revisions of "Element structure of special linear group:SL(2,5)"

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(Relationship with conjugacy class structure for an arbitrary special linear group of degree two)
(Conjugacy class structure)
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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  + 3x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math>  || 12 || 4 || 48 || No || No || Yes || <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}</math>
 
| 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  + 3x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math>  || 12 || 4 || 48 || No || No || Yes || <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}</math>
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| Diagonalizable over [[field:F25]], not over [[field:F5]]. Must necessarily have no repeated eigenvalues. || pair of square roots of <math>2</math> in [[field:F25]], pair of square roots of <math>3</math> in [[field:F25]] || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>x^2 - x + 1</math>, <math>x^2 + x  +1</math> || 20 || 2 || 40 || Yes ||  No || No || <math>\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}</math>, <math>\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}</math>
 
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| Diagonalizable over [[field:F5]] with ''distinct'' diagonal entries || <math>2,3</math> || <math>x^2 + 1</math> || <math>x^2 + 1</math>  || 30 || 1 || 30 || Yes|| Yes || No || <math>\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}</math>
 
| Diagonalizable over [[field:F5]] with ''distinct'' diagonal entries || <math>2,3</math> || <math>x^2 + 1</math> || <math>x^2 + 1</math>  || 30 || 1 || 30 || Yes|| Yes || No || <math>\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}</math>
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| Diagonalizable over [[field:F25]], not over [[field:F5]]. Must necessarily have no repeated eigenvalues. || pair of square roots of <math>2</math> in [[field:F25]], pair of square roots of <math>3</math> in [[field:F25]] || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>x^2 - x + 1</math>, <math>x^2 + x  +1</math> || 20 || 2 || 40 || Yes ||  No || No || <math>\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}</math>, <math>\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}</math>
 
 
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| Total || NA || NA || NA || NA || 9 || 120 || 72 || 32 || 48 || NA
 
| Total || NA || NA || NA || NA || 9 || 120 || 72 || 32 || 48 || NA
 
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Revision as of 00:35, 30 October 2010

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,5).
View element structure of particular groups | View other specific information about special linear group:SL(2,5)

This article gives detailed information about the element structure of special linear group:SL(2,5), which is a group of order 120.

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

Conjugacy class structure

Conjugacy classes

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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 \mathbb{F}_q? Splits in SL_2 relative to GL_2? Representative matrices (one per conjugacy class)
Scalar \{ 1, 1 \} or \{ -1,-1\} x^2 - 2x + 1 or x^2 + 2x + 1 x - 1 or x + 1 1 2 2 Yes Yes No \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} and \begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}
Not diagonal, Jordan block of size two \{ 1, 1 \} or \{ -1,-1\} x^2 - 2x + 1 or x^2  + 3x + 1 x^2 - 2x + 1 or x^2 + 2x + 1 12 4 48 No No Yes \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}, \begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. pair of square roots of 2 in field:F25, pair of square roots of 3 in field:F25 x^2 - x + 1, x^2 + x + 1 x^2 - x + 1, x^2 + x  +1 20 2 40 Yes No No \begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}, \begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}
Diagonalizable over field:F5 with distinct diagonal entries 2,3 x^2 + 1 x^2 + 1 30 1 30 Yes Yes No \begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}
Total NA NA NA NA 9 120 72 32 48 NA