# General affine group

## Definition

### In terms of dimension

Let be a natural number and be a field. The **general affine group** or **affine general linear group** of degree over , denoted , , , or , is defined as the external semidirect product of the vector space by the general linear group , acting by linear transformations.

While cannot be realized as a subgroup of , it *can* be realized as a subgroup of in a fairly typical way: the vector from is the first entries of the right column, the matrix from is the top left block, there is a in the bottom right corner, and zeroes elsewhere on the bottom row.

### In terms of vector spaces

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