Special affine group

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

Let $n$ be a natural number and $k$ be a field. The special affine group or affine special linear group of degree $n$ over $k$, denoted $SA(n,k)$, $SA_{n}(k)$, $ASL(n,k)$, or $ASL_{n}(k)$, is defined as the external semidirect product of the vector space $k^{n}$ by the special linear group $SL(n,k)$.

It can be viewed as a subgroup of the general affine group, which is the semidirect product of the vector space with the whole general linear group.

While the general affine group $GA(n,K)$ cannot be realized as a subgroup of the general linear group $GL(n,K)$, 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. Then $SA(n,K)$ is the subgroup of this group consisting of matrices with determinant $1$ (that is, the top left $n\times n$ block has determinant $1$.)