Element structure of special linear group:SL(2,7): Difference between revisions

Latest revision as of 20:45, 31 May 2012

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,7).
View element structure of particular groups | View other specific information about special linear group:SL(2,7)

This article gives detailed information about the element structure of special linear group:SL(2,7), which is a group of order 336.

Conjugacy class structure

Compare with element structure of special linear group of degree two over a finite field#Conjugacy class structure.

In the table below, we consider the group as $SL(2,q),q=7$ . The information is stated for generic odd $q$ and then computed numerically for $q=7$ .

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 ( $q=7$ )	Number of such conjugacy classes (generic odd $q$ )	Number of such conjugacy classes ( $q=7$ )	Total number of elements (generic odd $q$ )	Total number of elements ( $q=7$ )	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 $x^{2}+2x+1$	$(q^{2}-1)/2$	24	4	4	$2(q^{2}-1)$	96	[SHOW MORE] ${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$ , ${\begin{pmatrix}1&-1\\0&1\\\end{pmatrix}}$ , ${\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix}}$ , ${\begin{pmatrix}-1&-1\\0&-1\\\end{pmatrix}}$
Diagonalizable over $\mathbb {F} _{q^{2}}$ , i.e., field:F49, not over $\mathbb {F} _{q}$ , i.e., field:F7. Must necessarily have no repeated eigenvalues.	For $q=7$ : $\{{\sqrt {-1}},-{\sqrt {-1}}\}$ , $\{2+{\sqrt {3}},2-{\sqrt {3}}\}$ , $\{-2+{\sqrt {3}},-2-{\sqrt {3}}\}$	For $q=7$ : $x^{2}+1$ , $x^{2}-4x+1$ , $x^{2}-3x+1$	For $q=7$ : $x^{2}+1$ , $x^{2}-4x+1$ , $x^{2}-3x+1$	$q(q-1)$	42	$(q-1)/2$	3	$q(q-1)^{2}/2$	126	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
Diagonalizable over $\mathbb {F} _{q}$ , i.e., field:F7 with distinct diagonal entries	For $q=7$ : $\{2,4\}$ , $\{3,5\}$	For $q=7$ : $x^{2}-x+1$ , $x^{2}+x+1$	For $q=7$ : $x^{2}-x+1$ , $x^{2}+x+1$	$q(q+1)$	56	$(q-3)/2$	2	$q(q+1)(q-3)/2$	112	[SHOW MORE] ${\begin{pmatrix}2&0\\0&4\\\end{pmatrix}}$ , ${\begin{pmatrix}3&0\\0&5\\\end{pmatrix}}$
Total	NA	NA	NA	NA	NA	$q+4$	11	$q^{3}-q$	336	NA

@@ Line 8: / Line 8: @@
 See also [[element structure of special linear group of degree two]].
+==Conjugacy class structure==
+Compare with [[element structure of special linear group of degree two over a finite field#Conjugacy class structure]].
+In the table below, we consider the group as <math>SL(2,q), q = 7</math>. The information is stated for generic odd <math>q</math> and then computed numerically for <math>q = 7</math>.
 {| class="sortable" border="1"
-! Nature of conjugacy class  !! Eigenvalues !! Characteristic polynomial !! Minimal polynomial !! 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>?
+! 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 = 7</math>) !! Number of such conjugacy classes (generic odd <math>q</math>) !! Number of such conjugacy classes (<math>q = 7</math>) !! Total number of elements (generic odd <math>q</math>) !! Total number of elements (<math>q = 7</math>) !! Representative matrices (one per conjugacy class)
 |-
-| Diagonalizable over [[field:F7]] with ''distinct'' (and hence mutually inverse) diagonal entries || <math>\{ 2,4 \}</math> and <math>\{ 3,5 \}</math> || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || Same as characteristic polynomial  || 56 || 2 || 112 || Yes || Yes || No
+| 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>
 |-
-| Diagonalizable over [[field:F49]], not over [[field:F7]]. Must necessarily have no repeated eigenvalues. || Pair of conjugate elements of <math>\mathbb{F}_{49}</math> of norm 1 || <math>x^2 - 3x + 1</math>, <math>x^2 - 4x + 1</math> || Same as characteristic polynomial || 42 || 3 || 126 || Yes || No || No
+| 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> || 24 || 4 || 4 || <math>2(q^2 - 1)</math> || 96 || <toggledisplay><math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix}</math></toggledisplay>
 |-
-| Diagonalizable over <math>\mathbb{F}_q</math> with equal diagonal entries, hence a scalar || <math>\{ 1,1 \}</math> or <math>\{ -1,-1\}</math> || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || <math>x - a</math> where <math>a \in \{ -1,1\}</math> || 1 || 2 || 2 || Yes || Yes || No
+| Diagonalizable over <math>\mathbb{F}_{q^2}</math>, i.e., [[field:F49]], not over <math>\mathbb{F}_q</math>, i.e., [[field:F7]]. Must necessarily have no repeated eigenvalues. || For <math>q = 7</math>: <math>\{ \sqrt{-1}, -\sqrt{-1} \}</math>, <math>\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}</math>, <math>\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}</math> || For <math>q = 7</math>: <math>x^2 + 1</math>, <math>x^2 - 4x + 1</math>, <math>x^2 - 3x + 1</math> || For <math>q = 7</math>: <math>x^2 + 1</math>, <math>x^2 - 4x + 1</math>, <math>x^2 - 3x + 1</math> || <math>q(q - 1)</math> || 42 || <math>(q - 1)/2</math> || 3 || <math>q(q - 1)^2/2</math> || 126 || {{fillin}}
 |-
-| Not diagonal, has Jordan block of size two  || <math>1</math> (multiplicity 2) or <math>-1</math> (multiplicity 2) || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || <math>x - a</math> where <math>a \in \{ -1,1\}</math> || 24 || 4 || 96 || No || No || Yes (two conjugacy classes over <math>GL_2</math>, each splits into two over <math>SL_2</math>)
+| Diagonalizable over <math>\mathbb{F}_q</math>, i.e., [[field:F7]] with ''distinct'' diagonal entries || For <math>q = 7</math>: <math>\{ 2,4 \}</math>, <math>\{ 3,5 \}</math> || For <math>q = 7</math>: <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || For <math>q = 7</math>: <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>q(q+1)</math> || 56  || <math>(q - 3)/2</math> || 2 || <math>q(q+1)(q-3)/2</math> || 112 || <toggledisplay><math>\begin{pmatrix} 2 & 0 \\ 0 & 4 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 3 & 0 \\ 0 & 5 \\\end{pmatrix}</math></toggledisplay>
 |-
-| Total || NA || NA || NA || NA || 11 || 336 || 238 || 98 || 96
+! Total || NA || NA || NA || NA || NA || <math>q + 4</math> || 11 || <math>q^3 - q</math> || 336 || NA
 |}