Special affine group

In the mathematical study of transformation groups, the special affine group is the group of affine transformations of a fixed affine space which preserve volume. This is the affine analogue of the special linear group.

On Rⁿ, every special affine transformation has the form

$\mathbf{x}\mapsto A\mathbf{x}+\mathbf{b}$

where A is an n×n matrix of determinant one, and b is a fixed vector in Rⁿ. The special affine group is a subgroup of the affine group.