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In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
The affine group Aff(n, F) is an extension of GL(n, F) by the group of translations in F n. It can be written as a semidirect product: Aff(n, F) = GL(n, F) ⋉ F n. where GL(n, F) acts on F n in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space F n.
It is straightforward to generalize the notion of a residue-class-wise affine group to groups acting on suitable rings other than , though only little work in this direction has been done so far. See also the Collatz conjecture , which is an assertion about a surjective , but not injective residue-class-wise affine mapping.
The additive group: the affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its k {\displaystyle k} -points are isomorphic as a group to the additive group of k {\displaystyle k} ), and usually denoted by G a {\displaystyle \mathrm {G} _{a}} .
The affine group acts transitively on the points of an affine space, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, regular) action on these points; [14] indeed this can be used to give a definition of an affine space.
Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) on the Euclidean space E n. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought ...
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