General affine group
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 .