Projective special linear group of degree two: Difference between revisions

Revision as of 22:50, 14 May 2011

Definition

For a field or commutative unital ring

The projective special linear group of degree two over a field $k$ , or more generally over a commutative unital ring $R$ , is defined as the quotient of the special linear group of degree two over the same field or commutative unital ring by the subgroup of scalar matrices in that group. The group is denoted by $P S L (2, R)$ or $P S L_{2} (R)$ .

For a prime power

Suppose $q$ is a prime power. The projective special linear group $P S L (2, q)$ is defined as the projective special linear group of degree two over the field (unique up to isomorphism) with $q$ elements.

Particular cases

For prime powers $q$

Value of prime power $q$	Underlying prime $p$	Exponent on $p$ giving $q$	Group	Order	Second part of GAP ID (if applicable)	Comments
2	2	1	symmetric group:S3	6	1	not simple (one of two exceptions)
3	3	1	alternating group:A4	12	3	not simple (one of two exceptions)
4	2	2	alternating group:A5	60	5	minimal simple group
5	5	1	alternating group:A5	60	5	minimal simple group
7	7	1	projective special linear group:PSL(3,2)	168	42	minimal simple group
8	2	3	projective special linear group:PSL(2,8)	504	156	minimal simple group
9	3	2	alternating group:A6	360	114	minimal simple group
11	11	1	projective special linear group:PSL(2,11)	660	13	simple non-abelian group but not a minimal simple group -- contains alternating group:A5
13	13	1	projective special linear group:PSL(2,13)	1092	25	minimal simple group
16	2	4	projective special linear group:PSL(2,16)	4080	--	simple non-abelian group but not a minimal simple group -- contains alternating group:A5 as the subgroup $P S L (2, 4)$
17	17	1	projective special linear group:PSL(2,17)	2448	--	minimal simple group

Elements

Further information: element structure of projective special linear group of degree two

@@ Line 1: / Line 1: @@
 ==Definition==
+===For a field or commutative unital ring===
 The '''projective special linear group of degree two''' over a [[field]] <math>k</math>, or more generally over a [[commutative unital ring]] <math>R</math>, is defined as the quotient of the [[defining ingredient::special linear group of degree two]] over the same field or commutative unital ring by the subgroup of scalar matrices in that group. The group is denoted by <math>PSL(2,R)</math> or <math>PSL_2(R)</math>.
+===For a prime power===
+Suppose <math>q</matH> is a [[prime power]]. The projective special linear group <math>PSL(2,q)</math> is defined as the projective special linear group of degree two over the field (unique up to isomorphism) with <math>q</math> elements.
+==Particular cases==
+===For prime powers <math>q</math>===
+{| class="sortable" border="1"
+! Value of prime power <math>q</math> !! Underlying prime <math>p</math> !! Exponent on <math>p</math> giving <math>q</math> !! Group !! Order !! Second part of GAP ID (if applicable) !! Comments
+|-
+| 2 || 2 || 1 || [[symmetric group:S3]] || 6 || 1 || not simple (one of two exceptions)
+|-
+| 3 || 3 || 1 || [[alternating group:A4]] || 12 || 3 || not simple (one of two exceptions)
+|-
+| 4 || 2 || 2 || [[alternating group:A5]] || 60 || 5 || [[minimal simple group]]
+|-
+| 5 || 5 || 1 || [[alternating group:A5]] || 60 || 5 || [[minimal simple group]]
+|-
+| 7 || 7 || 1 || [[projective special linear group:PSL(3,2)]] || 168 || 42 || [[minimal simple group]]
+|-
+| 8 || 2 || 3 || [[projective special linear group:PSL(2,8)]] || 504 || 156 || [[minimal simple group]]
+|-
+| 9 || 3 || 2 || [[alternating group:A6]] || 360 || 114 || [[minimal simple group]]
+|-
+| 11 || 11 || 1 || [[projective special linear group:PSL(2,11)]] || 660 || 13 || [[simple non-abelian group]] but ''not'' a [[minimal simple group]] -- contains [[alternating group:A5]]
+|-
+| 13 || 13 || 1 || [[projective special linear group:PSL(2,13)]] || 1092 || 25 || [[minimal simple group]]
+|-
+| 16 || 2 || 4 || [[projective special linear group:PSL(2,16)]] || 4080 || -- || [[simple non-abelian group]] but ''not'' a [[minimal simple group]] -- contains [[alternating group:A5]] as the subgroup <math>PSL(2,4)</math>
+|-
+| 17 || 17 || 1 || [[projective special linear group:PSL(2,17)]] || 2448 || -- || [[minimal simple group]]
+|}
 ==Elements==
 {{further|[[element structure of projective special linear group of degree two]]}}