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.
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.
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 In the case that the affine group was constructed by starting with a vector space, the subgroup that stabilizes the origin (of the vector space) is the original GL(V).
Representing the affine group as a semidirect product of V by GL(V), then by construction of the semidirect product, the elements are pairs (v, M), where v is a vector in V and M is a linear transform in GL(V), and multiplication is given by This can be represented as the (n + 1) × (n + 1) block matrix where M is an n × n matrix over K, v an n × 1 column vector, 0 is a 1 × n row of zeros, and 1 is the 1 × 1 identity block matrix.
Formally, Aff(V) is naturally isomorphic to a subgroup of GL(V ⊕ K), with V embedded as the affine plane {(v, 1) | v ∈ V}, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of the) realization of this, with the n × n and 1 × 1 blocks corresponding to the direct sum decomposition V ⊕ K. A similar representation is any (n + 1) × (n + 1) matrix in which the entries in each column sum to 1.
The simplest paradigm may well be the case n = 1, that is, the upper triangular 2 × 2 matrices representing the affine group in one dimension.
Then compare with the order of Fp, we have hence χp = p − 1 is the dimension of the last irreducible representation.
Finally using the orthogonality of irreducible representations, we can complete the character table of Aff(Fp): The elements of
More precisely, given an affine transformation of an affine plane over the reals, an affine coordinate system exists on which it has one of the following forms, where a, b, and t are real numbers (the given conditions insure that transformations are invertible, but not for making the classes distinct; for example, the identity belongs to all the classes).
The transformations that do not preserve the orientation of the plane belong to cases 2 (with ab < 0) or 3 (with a < 0).
The proof may be done by first remarking that if an affine transformation has no fixed point, then the matrix of the associated linear map has an eigenvalue equal to one, and then using the Jordan normal form theorem for real matrices.
In terms of the semi-direct product, the special affine group consists of all pairs (M, v) with
where M is a linear transformation of whose determinant has absolute value 1 and v is any fixed translation vector.
of distance-preserving maps (isometries) of A is a subgroup of the affine group.
Geometrically, it is the subgroup of the affine group generated by the orthogonal reflections.