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)
(Relationship with conjugacy class structure for an arbitrary special linear group of degree two)
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===Relationship with conjugacy class structure for an arbitrary special linear group of degree two===
 
===Relationship with conjugacy class structure for an arbitrary special linear group of degree two===
  
{{further|[[element structure of special linear group of degree two]]}}
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{{further|[[element structure of special linear group of degree two over a finite field]]}}
 
 
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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)
 
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| Scalar || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 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  + 2x + 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>
 
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| Total || NA || NA || NA || NA || 9 || 120 || 72 || 32 || 48 || NA
 
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Revision as of 00:22, 19 February 2012

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

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 over a finite field

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 (generic odd q) Size of conjugacy class (q = 5) Number of such conjugacy classes (generic odd q) Number of such conjugacy classes (q = 5) Total number of elements (generic odd q) Total number of elements (q = 5) 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 1 2 2 2 2 \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 + 2x + 1 x^2 - 2x + 1 or x^2 + 2x + 1 (q^2 - 1)/2 12 4 4 2(q^2 - 1) 48 [SHOW MORE]
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. \{ 2 + \sqrt{3}, 2 - \sqrt{3} \} and \{ -2 + \sqrt{3}, -2 - \sqrt{3} \}, where \sqrt{3} is interpreted a an element of field:F25 that squares to 3 x^2 - x + 1, x^2 + x + 1 x^2 - x + 1, x^2 + x + 1 q(q - 1) 20 (q - 1)/2 2 q(q - 1)^2/2 40 \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 q(q+1) 30 (q - 3)/2 1 q(q+1)(q-3)/2 30 \begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}
Total NA NA NA NA NA q + 4 9 q^3 - q 120 NA