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.

Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars is the real or complex field, then the affine group is a Lie group.

Relation to general linear group

Construction from general linear group

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:

${\displaystyle \operatorname {Aff} (V)=V\rtimes \operatorname {GL} (V)}$

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:

${\displaystyle \operatorname {Aff} (n,K)=K^{n}\rtimes \operatorname {GL} (n,K)}$

where here the natural action of GL(n, K) on Kⁿ is matrix multiplication of a vector.

Stabilizer of a point

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 (A, p): 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

${\displaystyle 1\to V\to V\rtimes \operatorname {GL} (V)\to \operatorname {GL} (V)\to 1\,.}$

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).

Matrix representation

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

${\displaystyle (v,M)\cdot (w,N)=(v+Mw,MN)\,.}$

This can be represented as the (n + 1) × (n + 1) block matrix

${\displaystyle \left({\begin{array}{c|c}M&v\\\hline 0&1\end{array}}\right)}$

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.^[1] The similarity P for passing from the above kind to this kind is the (n + 1) × (n + 1) 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 n = 1, that is, the upper triangular 2 × 2 matrices representing the affine group in one dimension. It is a two-parameter non-Abelian Lie group, so with merely two generators (Lie algebra elements), A and B, such that = B, where

${\displaystyle A=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad B=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right)\,,}$

so that

${\displaystyle e^{aA+bB}=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)\,.}$

Character table of Aff(F_p)

Aff(F_p) has order p(p − 1). Since

${\displaystyle {\begin{pmatrix}c&d\\0&1\end{pmatrix}}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\begin{pmatrix}c&d\\0&1\end{pmatrix}}^{-1}={\begin{pmatrix}a&(1-a)d+bc\\0&1\end{pmatrix}}\,,}$

we know Aff(F_p) has p conjugacy classes, namely

${\displaystyle {\begin{aligned}C_{id}&=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}}\right\}\,,\\C_{1}&=\left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}^{*}\right\}\,,\\{\Bigg \{}C_{a}&=\left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}\right\}{\Bigg |}a\in \mathbf {F} _{p}\setminus \{0,1\}{\Bigg \}}\,.\end{aligned}}}$

Then we know that Aff(F_p) has p irreducible representations. By above paragraph (§ Matrix representation), there exist p − 1 one-dimensional representations, decided by the homomorphism

${\displaystyle {\mathrm{\rho <!--\; \rho \; -->}}_k:\mathrm{Aff}<!--\; \; -->({\mathbf{F}}_p)\to <!--\; \to \; -->{\mathbb{C}}^{}}$Zdroj:https://en.wikipedia.org?pojem=Special_affine_group
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