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

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| <math>\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}</math> || 1 || <math>\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}</math>
| <math>\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}</math> || 1 || <math>\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}</math>
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| <math>\begin{pmatrix}0 & -1\\ 1 & 0\\\end{pmatrix}</math> || 6 || <toggledisplay><math>\begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 0 & 1 \\ -1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 1 \\ 1 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 1 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ -1 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & -1 \\ -1 & 1 \\\end{pmatrix}</math></toggledisplay>
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| <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> || 4 || <toggledisplay><math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ -1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 0 \\ -1 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 0 & 1 \\ -1 & -1 \\\end{pmatrix}</math></toggledisplay>
| <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> || 4 || <toggledisplay><math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ -1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 0 \\ -1 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 0 & 1 \\ -1 & -1 \\\end{pmatrix}</math></toggledisplay>
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| <math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math> || 4 || <toggledisplay><math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ 1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 0 \\ 1 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 0 & -1 \\ 1 & 1 \\\end{pmatrix}</math></toggledisplay>
| <math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math> || 4 || <toggledisplay><math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ 1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 0 \\ 1 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 0 & -1 \\ 1 & 1 \\\end{pmatrix}</math></toggledisplay>
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| <math>\begin{pmatrix}0 & -1\\ 1 & 0\\\end{pmatrix}</math> || 6 || <toggledisplay><math>\begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 0 & 1 \\ -1 & 0 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 1 \\ 1 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 1 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ -1 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & -1 \\ -1 & 1 \\\end{pmatrix}</math></toggledisplay>
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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===

Revision as of 11:35, 18 July 2011

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 over a finite field.

Conjugacy class structure

Conjugacy classes

Note that since we are over field:F3, −1=2, so all the −1s below can be rewritten as 2s.

Conjugacy class representative Conjugacy class size List of all elements of conjugacy class
(1001) 1 (1001)
(−100−1) 1 (−100−1)
(1101) 4 [SHOW MORE]
(1−101) 4 [SHOW MORE]
(−110−1) 4 [SHOW MORE]
(−1−10−1) 4 [SHOW MORE]
(0−110) 6 [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 q) Size of conjugacy class (q=3) Number of such conjugacy classes (generic q) Number of such conjugacy classes (q=3) Total number of elements (generic q) Total number of elements (q=3) 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 1 2 2 2 2 (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 (q2−1)/2 4 4 4 2(q2−1) 16 [SHOW MORE]
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 q(q−1) 6 (q−1)/2 1 q(q−1)2/2 6 (0−110)
Diagonalizable over field:F3 with distinct diagonal entries -- -- -- q(q+1) 12 (q−3)/2 0 q(q+1)(q−3)/2 0 --
Total NA NA NA NA NA q+4 7 q3−q 24 NA
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