Outer linear group: Difference between revisions

Latest revision as of 19:51, 7 July 2019

Definition

In terms of the transpose-inverse map

The outer linear group of degree $n$ over a commutative unital ring $R$ is defined as the external semidirect product of the general linear group $GL(n,R)$ with a cyclic group of order two, where the non-identity element of the cyclic group acts by the transpose-inverse map.

Particular cases

Finite fields

Size of field	Degree (order of matrices)	Common name for the outer linear group	Order of group	Comment
$q$	1	Dihedral group $D_{2(q-1)}$	$2(q-1)$	Multiplicative group of field is cyclic of order $q-1$ , outer automorphism acts by inverse map.
$2^{n}$	2	Direct product of $SL(2,2^{n})$ and dihedral group $D_{2(2^{n}-1)}$
2	1	Cyclic group:Z2	$2$
2	2	Dihedral group:D12 (also, direct product of $S_{3}$ and $\mathbb {Z} _{2}$	$12=2^{2}\cdot 3$	supersolvable but not nilpotent.
3	2	Outer linear group:OL(2,3)	$96=2^{5}\cdot 3$	solvable
4	2	Direct product of A5 and S3	$360=2^{3}\cdot 3^{2}\cdot 5$
5	2	Outer linear group:OL(2,5)	$960=2^{6}\cdot 3\cdot 5$
2	3	Projective general linear group:PGL(2,7)	$336=2^{4}\cdot 3\cdot 7$

@@ Line 1: / Line 1: @@
+==Definition==
+===In terms of the transpose-inverse map===
+The '''outer linear group''' of degree <math>n</math> over a [[commutative unital ring]] <math>R</math> is defined as the [[external semidirect product]] of the [[defining ingredient::general linear group]] <math>GL(n,R)</math> with a [[cyclic group:Z2|cyclic group of order two]], where the non-identity element of the cyclic group acts by the [[defining ingredient::transpose-inverse map]].
 ==Particular cases==
 ===Finite fields===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-!Size of field !! Order of matrices !! Common name for the outer linear group !! Order of group !! Comment
+!Size of field !! Degree (order of matrices) !! Common name for the outer linear group !! Order of group !! Comment
 |-
-|<math>q</math> || 1 || Dihedral group <math>D_{2(q-1)}</math> || 2(q-1) || Multiplicative group of field is cyclic of order <math>q - 1</math>, outer automorphism acts by inverse map.
+|<math>q</math> || 1 || Dihedral group <math>D_{2(q-1)}</math> || <math>2(q-1)</math> || Multiplicative group of field is cyclic of order <math>q - 1</math>, outer automorphism acts by inverse map.
 |-
-|<math>2^n</math> || 2 || Direct product of <math>GL(2,2^n)</math> and [[cyclic group:Z2]] || ||
+|<math>2^n</math> || 2 || Direct product of <math>SL(2,2^n)</math> and dihedral group <math>D_{2(2^n - 1)}</math> || ||
 |-
 | 2 || 1 || [[Cyclic group:Z2]] || <math>2</math> ||
 |-
-| 2 || 2 || [[Direct product of S3 and Z2]] || <math>12 = 2^2 \cdot 3</math> || [[supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]].
+| 2 || 2 || [[Dihedral group:D12]] (also, direct product of <math>S_3</math> and <math>\mathbb{Z}_2</math> || <math>12 = 2^2 \cdot 3</math> || [[supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]].
 |-
-| 3 || 2 || ? || <math>96 = 2^5 \cdot 3</math> || [[solvable group|solvable]]
+| 3 || 2 || [[Outer linear group:OL(2,3)]] || <math>96 = 2^5 \cdot 3</math> || [[solvable group|solvable]]
 |-
-| 4 || 2 || [[Direct product of A5 and Z2]] || <math>120 = 2^3 \cdot 3 \cdot 5</math> ||
+| 4 || 2 || [[Direct product of A5 and S3]] || <math>360 = 2^3 \cdot 3^2 \cdot 5</math> ||
 |-
-| 5 || 2 || ? || <math>960 = 2^6 \cdot 3 \cdot 5</math> ||
+| 5 || 2 || [[Outer linear group:OL(2,5)]] || <math>960 = 2^6 \cdot 3 \cdot 5</math> ||
 |-
-| 2 || 3 || ? || <math>336 = 2^4 \cdot 3 \cdot 7</math> ||
+| 2 || 3 || [[Projective general linear group:PGL(2,7)]] || <math>336 = 2^4 \cdot 3 \cdot 7</math> ||
 |}