Special affine group

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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

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.