General affine 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 $GA(n,K)$, $GA_n(K)$, $AGL(n,K)$, or $AGL_n(K)$, is defined as the external semidirect product of the vector space $K^n$ by the 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 $GL(V)$.