General affine group

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 $$GA(n,K)$$, $$GA_n(K)$$, $$AGL(n,K)$$, or $$AGL_n(K)$$, is defined as the external semidirect product of the vector space $$K^n$$ by the defining ingredient::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 $$K^n$$ is the first $$n$$ entries of the right column, the matrix from $$GL(n,K)$$ 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 $$GA(V)$$, is defined as the external semidirect product of $$V$$ by $$GL(V)$$.