General affine group: Difference between revisions

Latest revision as of 11:21, 18 November 2023

Template:Field-parametrized linear algebraic group

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

In terms of dimension

Let $n$ be a natural number and $K$ be a field. The general affine group or affine general linear group of degree $n$ over $K$ , denoted $G A (n, K)$ , $G A_{n} (K)$ , $A G L (n, K)$ , or $A G L_{n} (K)$ , is defined as the external semidirect product of the vector space $K^{n}$ by the general linear group $G L (n, K)$ , acting by linear transformations.

While $G A (n, K)$ cannot be realized as a subgroup of $G L (n, K)$ , it can be realized as a subgroup of $G L (n + 1, K)$ in a fairly typical way: the vector from $K^{n}$ is the first $n$ entries of the right column, the matrix from $G L (n, K)$ is the top left $n \times n$ block, there is a $1$ in the bottom right corner, and zeroes elsewhere on the bottom row.

In terms of vector spaces

Let $V$ be a $K$ -vector space (which may be finite- or infinite-dimensional). The general affine group of $V$ , denoted $G A (V)$ , is defined as the external semidirect product of $V$ by $G L (V)$ .

Notation for general affine group over a finite field

For $q = p^{n}$ a prime power ( $p$ prime), we write $G A (n, q) = G A (n, F_{q})$ for the general affine group over the finite field with $q$ elements.

Particular cases

Finite general affine groups

Degree one

Further information: General affine group of degree one

$q$ (field size)	$p$ (underlying prime, field characteristic)	$G A (1, q)$	Order	Second part of GAP ID
2	2	cyclic group:Z2	2	1
3	3	symmetric group:S3	6	1
4	2	alternating group:A4	12	3
5	5	general affine group:GA(1,5)	20	3
7	7	general affine group:GA(1,7)	42	1
8	2	general affine group:GA(1,8)	56	11
9	3	general affine group:GA(1,9)	72	39

Infinite general affine groups

General affine group:GA(1,Q)

Linear representation theory

Further information: Linear representation theory of general affine group

Important subgroups

A particular subgroup of note is the special affine group.

@@ Line 1: / Line 1: @@
 {{field-parametrized linear algebraic group}}
+{{semibasicdef}}
 ==Definition==
@@ Line 5: / Line 6: @@
 ===In terms of dimension===
-Let <math>n</math> be a [[natural number]] and <math>k</math> be a [[field]]. The '''general affine group''' of order <math>n</math> over <math>k</math>, denoted <math>GA(n,k)</math> or <math>GA_n(k)</math>, is defined as the [[external semidirect product]] of the vector space <math>k^n</math> by the group <math>GL(n,k)</math>, acting by linear transformations.
+Let <math>n</math> be a [[natural number]] and <math>K</math> be a [[field]]. The '''general affine group''' or '''affine general linear group''' of degree <math>n</math> over <math>K</math>, denoted <math>GA(n,K)</math>, <math>GA_n(K)</math>, <math>AGL(n,K)</math>, or <math>AGL_n(K)</math>, is defined as the [[external semidirect product]] of the vector space <math>K^n</math> by the [[defining ingredient::general linear group]] <math>GL(n,K)</math>, acting by linear transformations.
-While <math>GA(n,k)</math> cannot be realized as a subgroup of <math>GL(n,k)</math>, it ''can'' be realized as a subgroup of <math>GL(n+1,k)</math> in a fairly typical way: the vector from <math>k^n</math> is the first <math>n</math> entries of the right column, the matrix from <math>GL(n,k)</math> is the top left <math>n \times n</math> block, there is a <math>1</math> in the bottom right corner, and zeroes elsewhere on the bottom row.
+While <math>GA(n,K)</math> cannot be realized as a subgroup of <math>GL(n,K)</math>, it ''can'' be realized as a subgroup of <math>GL(n+1,K)</math> in a fairly typical way: the vector from <math>K^n</math> is the first <math>n</math> entries of the right column, the matrix from <math>GL(n,K)</math> is the top left <math>n \times n</math> block, there is a <math>1</math> in the bottom right corner, and zeroes elsewhere on the bottom row.
 ===In terms of vector spaces===
-Let <math>V</math> be a <math>k</math>-vector space (which may be finite- or infinite-dimensional). The general affine group of <math>V</math>, denoted <math>GA(V)</math>, is defined as the external semidirect product of <math>V</math> by <math>GL(V)</math>.
+Let <math>V</math> be a <math>K</math>-vector space (which may be finite- or infinite-dimensional). The general affine group of <math>V</math>, denoted <math>GA(V)</math>, is defined as the external semidirect product of <math>V</math> by <math>GL(V)</math>.
+===Notation for general affine group over a finite field===
+For <math>q=p^n</math> a prime power (<math>p</math> prime), we write <math>GA(n, q) = GA(n, \mathbb{F}_q)</math> for the general affine group over the finite field with <math>q</math> elements.
+==Particular cases==
+===Finite general affine groups===
+====Degree one====
+{{further|[[General affine group of degree one]]}}
+{| class="sortable" border="1"
+! <math>q</math> (field size) !! <math>p</math> (underlying prime, field characteristic) !! <math>GA(1,q)</math> !! Order !! Second part of GAP ID
+|-
+| 2 || 2 || [[cyclic group:Z2]] || 2 || 1
+|-
+| 3 || 3 || [[symmetric group:S3]] || 6 || 1
+|-
+| 4 || 2 || [[alternating group:A4]] || 12 || 3
+|-
+| 5 || 5 || [[general affine group:GA(1,5)]] || 20 || 3
+|-
+| 7 || 7 || [[general affine group:GA(1,7)]] || 42 || 1
+|-
+| 8 || 2 || [[general affine group:GA(1,8)]] || 56 || 11
+|-
+| 9 || 3 || [[general affine group:GA(1,9)]] || 72 || 39
+|}
+===Infinite general affine groups===
+* [[General affine group:GA(1,Q)]]
+==Linear representation theory==
+{{further|[[Linear representation theory of general affine group]]}}
+==Important subgroups==
+A particular subgroup of note is the [[special affine group]].