Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.
It is a Lie group if K is the real or complex field or quaternions.
Concretely, given a vector space V, it has an underlying affine space A obtained by "forgetting" the origin, with V acting by translations, and the affine group of A can be described concretely as the semidirect product of V by GL(V), the general linear group of V:
The action of GL(V) on V is the natural one (linear transformations are automorphisms), so this defines a semidirect product.
In terms of matrices, one writes:
where here the natural action of GL(n,K) on Kn is matrix multiplication of a vector.
Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in Aff(2,R) is isomorphic to GL(2,R)); formally, it is the general linear group of the vector space : recall that if one fixes a point, an affine space becomes a vector space.
All these subgroups are conjugate, where conjugation is given by translation from p to q (which is uniquely defined), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the short exact sequence
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