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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 GA(n,k), GAn(k), AGL(n,k), or AGLn(k), is defined as the external semidirect product of the vector space kn by the general linear group GL(n,k), acting by linear transformations.
While GA(n,k) cannot be realized as a subgroup of GL(n,k), it can be realized as a subgroup of GL(n + 1,k) in a fairly typical way: the vector from kn is the first n entries of the right column, the matrix from GL(n,k) is the top left 
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 GA(V), is defined as the external semidirect product of V by GL(V).
