General affine group: Difference between revisions

Revision as of 19:59, 17 November 2023

Template:Field-parametrized linear algebraic group

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.

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 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.