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

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

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 $F_{q}$ ?	Splits in $S L_{2}$ relative to $G L_{2}$ ?	Representative matrices (one per conjugacy class)
Scalar	${1, 1}$ or ${- 1, - 1}$	$x^{2} - 2 x + 1$ or $x^{2} + 2 x + 1$	$x - 1$ or $x + 1$	1	2	2	Yes	Yes	No	$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ and $(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})$
Not diagonal, Jordan block of size two	${1, 1}$ or ${- 1, - 1}$	$x^{2} - 2 x + 1$ or $x^{2} + 2 x + 1$	$x^{2} - 2 x + 1$ or $x^{2} + 2 x + 1$	12	4	48	No	No	Yes	$(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ , $(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix})$ , $(\begin{matrix} - 1 & 1 \\ 0 & - 1 \end{matrix})$ , $(\begin{matrix} - 1 & 2 \\ 0 & - 1 \end{matrix})$
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	$(\begin{matrix} 0 & - 1 \\ 1 & 1 \end{matrix})$ , $(\begin{matrix} 0 & - 1 \\ 1 & - 1 \end{matrix})$
Diagonalizable over field:F5 with distinct diagonal entries	$2, 3$	$x^{2} + 1$	$x^{2} + 1$	30	1	30	Yes	Yes	No	$(\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix})$
Total	NA	NA	NA	NA	9	120	72	32	48	NA

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 ${- 1, - 1}$	$x^{2} - 2 x + 1$ or $x^{2} + 2 x + 1$	$x - 1$ or $x + 1$	1	1	2	2	2	2	$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ and $(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})$
Not diagonal, Jordan block of size two	${1, 1}$ or ${- 1, - 1}$	$x^{2} - 2 x + 1$ or $x^{2} + 2 x + 1$	$x^{2} - 2 x + 1$ or $x^{2} + 2 x + 1$	$(q^{2} - 1) / 2$	12	4	4	$2 (q^{2} - 1)$	48	[SHOW MORE] $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ , $(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix})$ , $(\begin{matrix} - 1 & 1 \\ 0 & - 1 \end{matrix})$ , $(\begin{matrix} - 1 & 2 \\ 0 & - 1 \end{matrix})$
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	$(\begin{matrix} 0 & - 1 \\ 1 & 1 \end{matrix})$ , $(\begin{matrix} 0 & - 1 \\ 1 & - 1 \end{matrix})$
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	$(\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix})$
Total	NA	NA	NA	NA	NA	$q + 4$	9	$q^{3} - q$	120	NA

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 | Total || NA || NA || NA || NA || 9 || 120 || 72 || 32 || 48 || NA
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+{| class="sortable" border="1"
+! 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 <math>q</math>) !! Size of conjugacy class (<math>q = 5</math>) !! Number of such conjugacy classes (generic odd <math>q</math>) !! Number of such conjugacy classes (<math>q = 5</math>) !! Total number of elements (generic odd <math>q</math>) !! Total number of elements (<math>q = 5</math>) !! Representative matrices (one per conjugacy class)
+|-
+| 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 || 1 || 2 || 2 || 2 || 2 || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} -1 & 0 \\ 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 + 2x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math>  || <math>(q^2 - 1)/2</math> || 12 || 4 || 4 || <math>2(q^2 - 1)</math> || 48 || <toggledisplay><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></toggledisplay>
+|-
+| Diagonalizable over [[field:F25]], not over [[field:F5]]. Must necessarily have no repeated eigenvalues. || <math>\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}</math> and <math>\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}</math>, where <math>\sqrt{3}</math> is interpreted a an element of [[field:F25]] that squares to 3 || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>q(q - 1)</math> || 20 || <math>(q - 1)/2</math> || 2 || <math>q(q - 1)^2/2</math> || 40 || <math>\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}</math>, <math>\begin{pmatrix}0 & -1\\ 1 & -1\\\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> || <math>q(q+1)</math> || 30 || <math>(q - 3)/2</math> || 1 || <math>q(q+1)(q-3)/2</math> ||30 || <math>\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}</math>
+|-
+| Total || NA || NA || NA || NA || NA || <math>q + 4</math> || 9 || <math>q^3 - q</math> || 120 || NA
 |}