General affine group

From Groupprops

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 $G L (V)$ .