Difference between revisions of "Element structure of special linear group:SL(2,5)"
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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> | | 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 + | + | | 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> | | 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> |
Revision as of 21:35, 18 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
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 ![]() |
Splits in ![]() ![]() |
Representative matrices (one per conjugacy class) |
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Scalar | ![]() ![]() |
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1 | 2 | 2 | Yes | Yes | No | ![]() ![]() |
Not diagonal, Jordan block of size two | ![]() ![]() |
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12 | 4 | 48 | No | No | Yes | ![]() ![]() ![]() ![]() |
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. | pair of square roots of ![]() ![]() |
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20 | 2 | 40 | Yes | No | No | ![]() ![]() |
Diagonalizable over field:F5 with distinct diagonal entries | ![]() |
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30 | 1 | 30 | Yes | Yes | No | ![]() |
Total | NA | NA | NA | NA | 9 | 120 | 72 | 32 | 48 | NA |