Inmathematics, theaffine group orgeneral affine group of anyaffine space is thegroup of all invertibleaffine transformations from the space into itself. In the case of aEuclidean space (where the associated field of scalars is thereal 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 aLie group.

Concretely, given a vector spaceV, it has an underlyingaffine spaceA obtained by "forgetting" the origin, withV acting by translations, and the affine group ofA can be described concretely as thesemidirect product ofV byGL(V), thegeneral linear group ofV:

${\displaystyle \mathrm{Aff}(V)=V⋊\mathrm{GL}(V)}$

The action ofGL(V) onV is the natural one (linear transformations are automorphisms), so this defines asemidirect product.

In terms of matrices, one writes:

${\displaystyle \mathrm{Aff}(n,K)=K^n⋊\mathrm{GL}(n,K)}$

where here the natural action ofGL(n,K) onKⁿ ismatrix multiplication of a vector.

Given the affine group of an affine spaceA, thestabilizer of a pointp is isomorphic to the general linear group of the same dimension (so the stabilizer of a point inAff(2,R) is isomorphic toGL(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 avector space.

All these subgroups are conjugate, where conjugation is given by translation fromp toq (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 theshort exact sequence

${\displaystyle 1\to V\to V⋊\mathrm{GL}(V)\to \mathrm{GL}(V)\to 1.}$

In the case that the affine group was constructed bystarting with a vector space, the subgroup that stabilizes the origin (of the vector space) is the originalGL(V).

Representing the affine group as a semidirect product ofV byGL(V), thenby construction of the semidirect product, the elements are pairs(v,M), wherev is a vector inV andM is a linear transform inGL(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

whereM is ann ×n matrix overK,v ann × 1 column vector, 0 is a1 ×n row of zeros, and 1 is the1 × 1 identity block matrix.

Formally,Aff(V) is naturally isomorphic to a subgroup ofGL(V ⊕K), withV 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 then ×n and1 × 1 blocks corresponding to the direct sum decompositionV ⊕K.

Asimilar representation is any(n + 1) × (n + 1) matrix in which the entries in each column sum to 1.^[1] ThesimilarityP 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 casen = 1, that is, the upper triangular2 × 2 matrices representing the affine group in one dimension. It is a two-parameternon-AbelianLie group, so with merely two generators (Lie algebra elements),A andB, such that[A,B] =B, where

so that

Aff(F_p) has orderp(p − 1). Since

we knowAff(F_p) hasp conjugacy classes, namely

Then we know thatAff(F_p) hasp irreducible representations. By above paragraph (§ Matrix representation), there existp − 1 one-dimensional representations, decided by the homomorphism

${\displaystyle {\rho }_k:\mathrm{Aff}({\mathbf{F}}_p)\to {\mathbb{C}}^{\ast }}$

fork = 1, 2,…p − 1, where

andi² = −1,a =g^j,g is a generator of the groupF^∗
_p. Then compare with the order ofF_p, we have

${\displaystyle p(p-1)=p-1+{\chi }_p^2,}$

henceχ_p =p − 1 is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table ofAff(F_p):

The elements of${\displaystyle \mathrm{Aff}(2,\mathbb{R})}$ can take a simple form on a well-chosenaffine coordinate system. More precisely, given an affine transformation of anaffine plane over thereals, an affine coordinate system exists on which it has one of the following forms, wherea,b, andt 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).

Case 1 corresponds totranslations.

Case 2 corresponds toscalings that may differ in two different directions. When working with aEuclidean plane these directions need not beperpendicular, since thecoordinate axes need not be perpendicular.

Case 3 corresponds to a scaling in one direction and a translation in another one.

Case 4 corresponds to ashear mapping combined with a dilation.

Case 5 corresponds to ashear mapping combined with a dilation.

Case 6 corresponds tosimilarities when the coordinate axes are perpendicular.

The affine transformations without anyfixed point belong to cases 1, 3, and 5. The transformations that do not preserve the orientation of the plane belong to cases 2 (withab < 0) or 3 (witha < 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 aneigenvalue equal to one, and then using theJordan normal form theorem for real matrices.

Given any subgroupG < GL(V) of thegeneral linear group, one can produce an affine group, sometimes denotedAff(G), analogously asAff(G) :=V ⋊G.

More generally and abstractly, given any groupG and arepresentation${\displaystyle \rho :G\to \mathrm{GL}(V)}$ ofG on a vector spaceV, one gets^{[note 1]} an associated affine groupV ⋊_ρG: one can say that the affine group obtained is "agroup extension by a vector representation", and, as above, one has the short exact sequence$${\displaystyle 1\to V\to V{⋊}_{\rho }G\to G\to 1.}$$

The subset of all invertible affine transformations that preserve a fixedvolume form up to sign is called thespecial affine group. (The transformations themselves are sometimes calledequiaffinities.) This group is the affine analogue of thespecial linear group. In terms of the semi-direct product, the special affine group consists of all pairs(M,v) with${\displaystyle |det(M)|=1}$, that is, the affine transformations$${\displaystyle x\mapsto Mx+v}$$whereM is a linear transformation of whose determinant has absolute value 1 andv is any fixed translation vector.^[2][3]

The subgroup of the special affine group consisting of those transformations whose linear part has determinant 1 is the group of orientation- and volume-preserving maps. Algebraically, this group is a semidirect product${\displaystyle SL(V)⋉V}$ of the special linear group of${\displaystyle V}$ with the translations. It is generated by theshear mappings.

Presuming knowledge ofprojectivity and the projective group ofprojective geometry, the affine group can be easily specified. For example, Günter Ewald wrote:^[4]

The set${\displaystyle \mathfrak{P}}$ of all projective collineations ofPⁿ is a group which we may call theprojective group ofPⁿ. If we proceed fromPⁿ to the affine spaceAⁿ by declaring ahyperplaneω to be ahyperplane at infinity, we obtain theaffine group${\displaystyle \mathfrak{A}}$ ofAⁿ as thesubgroup of${\displaystyle \mathfrak{P}}$ consisting of all elements of${\displaystyle \mathfrak{P}}$ that leaveω fixed.

${\displaystyle \mathfrak{A}\subset \mathfrak{P}}$

When the affine spaceA is a Euclidean space (over the field of real numbers), the group${\displaystyle \mathcal{E}}$ of distance-preserving maps (isometries) ofA is a subgroup of the affine group. Algebraically, this group is a semidirect product${\displaystyle O(V)⋉V}$ of theorthogonal group of${\displaystyle V}$ with the translations. Geometrically, it is the subgroup of the affine group generated by the orthogonal reflections.

ThePoincaré group is the affine group of theLorentz groupO(1,3):

${\displaystyle {\mathbf{R}}^{1,3}⋊\mathrm{O}(1,3)}$

This example is very important inrelativity.

• Affine Coxeter group – certain discrete subgroups of the affine group on aEuclidean space that preserve alattice
• Holomorph

• Affine geometry
• Group theory
• Lie groups

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