Concretely, given a vector space, it has an underlying affine space obtained by "forgetting" the origin, with acting by translations, and the affine group of can be described concretely as the semidirect product of by, the general linear group of : The action of on is the natural one, so this defines a semidirect product. In terms of matrices, one writes: where here the natural action of on is matrix multiplication of a vector.
Stabilizer of a point
Given the affine group of an affine space, the stabilizer of a point is isomorphic to the general linear group of the same dimension ; 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 to , 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 In the case that the affine group was constructed by starting with a vector space, the subgroup that stabilizes the origin is the original.
Representing the affine group as a semidirect product of by, then by construction of the semidirect product, the elements are pairs, where is a vector in and is a linear transform in, and multiplication is given by: This can be represented as the block matrix: where is an matrix over, an column vector, 0 is a row of zeros, and 1 is the identity block matrix. Formally, is naturally isomorphic to a subgroup of, with embedded as the affine plane, namely the stabilizer of this affine plane; the above matrix formulation is the blocks corresponding to the direct sum decomposition. A similar representation is any matrix in which the entries in each column sum to 1. The similarity for passing from the above kind to this kind is the identity matrix with the bottom row replaced by a row of all ones. Each of these two classes of matrices is closed under matrix multiplication. The simplest paradigm may well be the case, that is, the upper triangular matrices representing the affine group in one dimension. It is a two-parameter non-Abelian Lie group, so with merely two generators, and, such that, where so that
has order. Since we know has conjugacy classes, namely Then we know that has irreducible representations. By above paragraph, there exist one-dimensional representations, decided by the homomorphism for, where and,, is a generator of the group. Then compare with the order of, we have hence is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of :
Planar affine group
According to Rafael Artzy, "The linear part of each affinity can be brought into one of the following standard forms by a coordinate transformation followed by a dilation from the origin: where the coefficients,,, and are real numbers." Case 1 corresponds to similarity transformations which generate a subgroup of similarities. Euclidean geometry corresponds to the subgroup of congruences. It is characterized by Euclidean distance or angle, which are invariant under the subgroup of rotations. Case 2 corresponds to shear mappings. An important application is absolute time and space where Galilean transformations relate frames of reference. They generate the Galilean group. Case 3 corresponds to squeeze mapping. These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane. The geometry associated with this group is characterized by hyperbolic angle, which is a measure that is invariant under the subgroup of squeeze mappings. Using the above matrix representation of the affine group on the plane, the matrix is a 2 × 2 real matrix. Accordingly, a non-singular must have one of three forms that correspond to the trichotomy of Artzy.
Other affine groups
General case
Given any subgroup of the general linear group, one can produce an affine group, sometimes denoted analogously as. More generally and abstractly, given any group and a representation of on a vector space, one gets an associated affine group : one can say that the affine group obtained is "a group extension by a vector representation", and as above, one has the short exact sequence:
Special affine group
The subset of all invertible affine transformations preserving a fixed volume form, or in terms of the semi-direct product, the set of all elements with of determinant 1, is a subgroup known as the special affine group.
