List of Transformation Groups

Introduction

This article lists the transformation groups of the corpus — the groups that are realised as the transformations of a set carrying a structure — together with the structure each preserves and the article that introduces it. The umbrella notion is that of Transformation Groups: a group acts on a set $X$ through a homomorphism $\rho : G \to \operatorname{Sym}(X)$, the action is faithful when that homomorphism is injective, and an automorphism group $\operatorname{Aut}(X)$ is the subgroup of $\operatorname{Sym}(X)$ preserving whatever structure $X$ carries. The list is the entry point for the families that the rest of the category develops: the linear, classical, projective, affine, Euclidean, isometry, Möbius, conformal, symplectic, discrete and automorphism families, each of which is a catalogue of its own.

Every entry points to the article that introduces the group or the action. The article introduces nothing and proves nothing: it records what each group preserves, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the groups that act faithfully on a space it lists the actions that fail faithfulness — the action of $GL_n$ on the projective space, whose kernel is the scalars — the isometry group of a general Riemannian manifold, which is too small for the manifold to be a homogeneous space, the structure group of a bundle, which acts on the fibres and not on the base, and the mapping class group, which acts on the curve complex rather than on the surface by a faithful action of homeomorphisms, each with the failure named and the article that records it. The Erlangen program itself is the subject of Transformation Groups and the Erlangen Program, and the general theory is taken from Transformation Groups.

The Transformations of a Set and the Hierarchy of Automorphisms

The hierarchy $\operatorname{Sym}(X) \supset \operatorname{Aut}(X, \text{structure}) \supset \cdots$ records that a transformation group is determined by what it is required to preserve: each layer of structure cuts down the group.

Object	The property it has	Introduced in
the symmetric group $\operatorname{Sym}(X)$	the group of all bijections of a bare set; the largest transformation group of $X$	Transformation Groups
the symmetric group $S_n$	$\operatorname{Sym}(\{1,\ldots,n\})$, of order $n!$	Transformation Groups; Groups
an action $G \times X \to X$, $a\cdot x$	a homomorphism $\rho : G \to \operatorname{Sym}(X)$; the realisation of an abstract group by transformations	Transformation Groups; Group Actions and Structure
a faithful action	$\ker\rho = 1$; the group is a subgroup of $\operatorname{Sym}(X)$	Transformation Groups
the orbit and the stabiliser	$\operatorname{Orb}(x)$ and $\operatorname{Stab}(x)$, with the orbit–stabiliser theorem and $G/H$ for a transitive action	Group Actions and Structure; Transformation Groups
Cayley's embedding $G \hookrightarrow \operatorname{Sym}(G)$	the left regular action; every group is a transformation group of itself	Transformation Groups
the automorphism group $\operatorname{Aut}(X)$	the transformations preserving the structure on $X$; $\operatorname{Aut}(X) = \operatorname{End}(X)^\times$	Transformation Groups
the inner automorphisms $\operatorname{Inn}(G)$	the conjugations $x \mapsto gxg^{-1}$; $\operatorname{Inn}(G) \cong G/Z(G)$	Transformation Groups; List of Automorphism Groups
the automorphism group $\operatorname{Aut}(G)$ of a group	the transformations preserving the group operation, with $\operatorname{Out}(G) = \operatorname{Aut}(G)/\operatorname{Inn}(G)$	List of Automorphism Groups
the transformation group of a ring $\operatorname{Aut}(R)$	the automorphisms preserving addition and multiplication; the Frobenius in prime characteristic	Ring and Field Automorphisms; List of Automorphism Groups
the transformation group of an algebra $\operatorname{Aut}_R(A)$	the $R$-algebra automorphisms; $\operatorname{Der}_R(A)$ the infinitesimal ones	Automorphisms and Derivations of Algebras
the transformation group of a module $\operatorname{Aut}_A(M)$	the $A$-linear automorphisms; $\operatorname{End}_A(M)$ the monoid	Automorphisms of Modules over an Algebra

The Families of Transformation Groups

Each family is the transformation group of a space with a definite structure, and each is developed in a catalogue or an article of its own. The table is the entry point of the category.

Family	The structure preserved	Introduced in
the linear groups $GL(V)$, $SL(V)$	the linear structure of a vector space	The General Linear Group; List of Linear Geometric Groups
the classical groups $O(V,Q)$, $U(V,h)$, $Sp(V,\omega)$	a quadratic, Hermitian or symplectic form	Matrix Groups and Classical Groups; List of Classical Geometric Groups
the projective groups $PGL(V)$, $PSL(V)$, $PO$, $PSU$, $PSp$	the incidence structure of a projective space; the quotients by the centre	Projective Geometry; List of Projective Geometric Groups
the affine group $\operatorname{Aff}(n) = \mathbb{R}^n \rtimes GL(n)$	the affine structure; the automorphism group of affine space	Affine Spaces and Translations; List of Affine and Euclidean Groups
the Euclidean group $E(n) = \mathbb{R}^n \rtimes O(n)$	the Euclidean distance and its isometries	Euclidean Geometry; List of Affine and Euclidean Groups; List of Isometry and Symmetry Groups
the similarity group $\operatorname{Sim}(n)$	the distance up to a scalar; translations, rotations, reflections and dilatations	Euclidean Geometry; List of Affine and Euclidean Groups
the isometry groups $\operatorname{Isom}(\mathbb{R}^n)$, $\operatorname{Isom}(\mathbb{H}^n)$, $\operatorname{Isom}(S^n)$	a metric; the model spaces of constant curvature	Non-Euclidean Geometry; List of Isometry and Symmetry Groups
the Möbius and conformal groups $\operatorname{Möb}(n)$, $\operatorname{Conf}(M)$	a conformal structure; the Möbius group generated by inversions	Möbius and Lie Sphere Geometry; Conformal Geometry; List of Möbius and Conformal Groups
the symplectic group $Sp(2n,\mathbb{R})$ and the symplectomorphisms	a symplectic form, and the group of diffeomorphisms preserving it	Symplectic Geometry; List of Symplectic Geometries
the orthogonal group of a quadratic space	a quadratic form, through the reflection theorem of Cartan–Dieudonné	Isometries and Orthogonal Transformations
the discrete geometric groups	a lattice or a crystallographic structure; discreteness and cocompactness	Symmetry, Point and Crystallographic Groups; List of Discrete Geometric Groups
the homeomorphism and diffeomorphism groups	a topological or smooth structure; infinite-dimensional transformation groups	Diffeomorphism Groups; List of Diffeomorphism and Homeomorphism Groups
the mapping class group $\operatorname{Mod}(S)$	the isotopy classes of homeomorphisms; it acts on the curve complex	Mapping Class Groups
the birational group $\operatorname{Bir}(X)$	the birational equivalence of a variety	Algebraic Geometry
a Lie group acting on itself	the left translations; the group as a homogeneous space of itself	Lie Groups; Homogeneous Spaces
the Thompson groups $F$, $T$, $V$	the piecewise-linear homeomorphisms of the interval, the circle and the Cantor set	Thompson Groups and the Cantor Set
the Galois group $\operatorname{Gal}(L/K)$	the field automorphisms fixing $K$; the transformation group of a field extension	Galois Theory; List of Automorphism Groups

The Erlangen Program and the Klein Geometries

Klein's principle reads a geometry as the study of the invariants of a transformation group, and a geometry so presented is a pair $(G,H)$ with $G$ a Lie group, $H$ a closed subgroup and $G/H$ the homogeneous space. The program and its limits are the subject of Transformation Groups and the Erlangen Program, and the geometries it organises are catalogued separately.

Object or principle	The statement or the structure	Introduced in
Klein's principle	a geometry is the study of the invariants of a transformation group; the correspondence $G \to \operatorname{Aut}(X)$	Transformation Groups and the Erlangen Program
a Klein geometry $(G,H)$	a Lie group and a closed subgroup, with the homogeneous space $G/H$ as the model space	List of Klein Geometries; Homogeneous Spaces
homogeneous spaces	$G/H$ with the smooth structure making $G \to G/H$ a submersion, of dimension $\dim G - \dim H$	Homogeneous Spaces; Lie Groups
Euclidean, affine, projective geometry	the geometries of the corresponding groups $E(n)$, $\operatorname{Aff}(n)$, $PGL(n+1)$	List of Erlangen Program Geometries; List of Klein Geometries
hyperbolic, spherical geometry	the geometries of $O(n,1)$ and $O(n+1)$; the space forms	List of Erlangen Program Geometries; List of Riemannian Geometries
conformal and Möbius geometry	the geometries of $O(n+1,1)$ and of the Möbius group	List of Möbius and Conformal Groups; List of Erlangen Program Geometries
symplectic geometry	the geometry of the symplectic group and the symplectic form	List of Symplectic Geometries; List of Erlangen Program Geometries
the limits of the program	a Riemannian manifold of general curvature is not a Klein geometry; its isometry group need not act transitively	Transformation Groups and the Erlangen Program

The Constructions on an Action

An action is a homomorphism $G \to \operatorname{Sym}(X)$, and the constructions of the group theory of actions are the transformations of the group into $\operatorname{Sym}(X)$ and of the set into its orbits.

Object or construction	The property it has	Introduced in
the action on the cosets $G/H$	the transitive $G$-set of the left cosets; every transitive $G$-set is of this form	Group Actions and Structure; Transformation Groups
the conjugation action $G$ on itself	$g \cdot x = gxg^{-1}$; its orbits are the conjugacy classes and its fixed points the centre	Group Actions and Structure
the fixed-point set $\operatorname{Fix}(a)$	the points fixed by $a$; the fixed-point formula of Burnside's lemma counts them	Group Actions and Structure
the double coset $HaK$	the orbits of $H$ on $G/K$; the decomposition that generalises the coset decomposition	Group Actions and Structure
the class equation	$\lvert G\rvert = \lvert Z(G)\rvert + \sum_i [G : C_G(x_i)]$; the counting identity of the conjugation action	Group Actions and Structure
the restriction of an action to a subgroup	$G$ acting on $X$ restricts to $H \leq G$; the orbits refine	Group Actions and Structure; Transformation Groups
the product action on $X \times Y$	$g\cdot(x,y) = (gx, gy)$; the action of a direct product $G_1 \times G_2$	Group Actions and Structure
the induced action on the powers	the action of $G$ on the subsets, on the $k$-subsets and on the functions of a $G$-set	Group Actions and Structure
Burnside's lemma	the number of orbits is the average number of fixed points, $\lvert X/G\rvert = \tfrac1{\lvert G\rvert}\sum_g \lvert\operatorname{Fix}(g)\rvert$	Group Actions and Structure

Non-examples and Warnings

Object	Why the expected statement fails	Introduced in
the action of $GL_n(K)$ on $P^{n-1}(K)$	it is not faithful; the kernel is the group of scalars, and the faithful group is the projective quotient $PGL_n(K)$	Projective Geometry; List of Projective Geometric Groups
the trivial action of a group on a set	it is not faithful, except for the trivial group; $\ker\rho = G$	Group Actions and Structure
the isometry group of a general Riemannian manifold	it need not act transitively, so the manifold is not a homogeneous space; the Erlangen principle does not apply	Transformation Groups and the Erlangen Program
the structure group of a principal bundle	it acts on the fibres of the bundle and not on the base; it is a transformation group of the total space only through the local trivialisations	Fibre Bundles, Connections and Curvature
the mapping class group of a surface	it does not act by homeomorphisms on the surface itself; it acts faithfully on the set of isotopy classes of simple closed curves	Mapping Class Groups
an abstract group with no faithful action registered	a group is a transformation group only through an action; the correspondence $G \to \operatorname{Aut}(X)$ requires the action to be named	Transformation Groups
the symplectomorphism group of a symplectic manifold	it is infinite-dimensional and not the finite-dimensional symplectic group $Sp(2n,\mathbb{R})$	Symplectic Geometry

Objects that a reader may expect in a list of transformation groups, and does not find here.

Object	Why it is not listed	Introduced in
$O(V,Q)$, $U(V,h)$, $Sp(V,\omega)$ as groups	they are defined by a form, so they are introduced among the classical geometric groups and are listed there	List of Classical Geometric Groups
the spin and pin groups	they are constructed from the Clifford algebra and are listed with it	List of Clifford Algebras and Spin Groups
the infinite-dimensional Lie groups	recorded with Hilbert's fifth problem; they are transformation groups without a finite dimension	Hilbert's Fifth Problem and Infinite-Dimensional Lie Theory
the absolute Galois group	it acts on the algebraic closure; it is not the transformation group of a finite-dimensional geometry	Galois Cohomology; List of Automorphism Groups
the $p$-adic algebraic groups	they act on buildings and on $p$-adic symmetric spaces; recorded among the topological groups	List of Topological Groups; Buildings and Tits Systems

Summary

This article has listed the transformation groups of the corpus: the transformations of a bare set and the hierarchy $\operatorname{Sym}(X) \supset \operatorname{Aut}(X)$, with the actions, orbits, stabilisers and automorphism groups of a group, a ring, an algebra and a module; the families in which the category is developed — the linear, classical, projective, affine, Euclidean, isometry, Möbius, conformal, symplectic, discrete, homeomorphism and automorphism families, each with the structure it preserves; and the Erlangen program, which reads a geometry as the invariants of a transformation group and presents it as a Klein geometry $G/H$. Beside the examples stand the non-examples: the unfaithful action on projective space, the trivial action, the isometry group of a general Riemannian manifold, the structure group of a bundle, and the mapping class group acting on the curve complex. The list introduces and proves nothing; it is the entry point of the geometric-group family of the corpus.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are those of the introducing articles.

Symbol	Meaning
$\operatorname{Sym}(X)$, $S_n$	the symmetric group of a set; the symmetric group on $n$ letters
$\operatorname{Aut}(X)$, $\operatorname{End}(X)$	the automorphism group and the endomorphism monoid of a structure
$\rho : G \to \operatorname{Sym}(X)$	the permutation representation of an action; $\ker\rho = 1$ when faithful
$\operatorname{Orb}(x)$, $\operatorname{Stab}(x)$, $G/H$	orbit, stabiliser and coset space of a transitive action
$\operatorname{Aut}(G)$, $\operatorname{Inn}(G)$, $\operatorname{Out}(G)$	the automorphism, inner and outer groups of a group
$E(n)$, $\operatorname{Aff}(n)$, $\operatorname{Sim}(n)$	the Euclidean, affine and similarity groups
$GL(V)$, $SL(V)$, $PGL(V)$, $PSL(V)$	the linear and projective linear groups
$O(V,Q)$, $U(V,h)$, $Sp(V,\omega)$	the classical groups of a quadratic, Hermitian and symplectic form
$\operatorname{Möb}(n)$, $\operatorname{Conf}(M)$	the Möbius group and the conformal group
$\operatorname{Isom}(X)$, $\operatorname{Diff}(M)$, $\operatorname{Homeo}(X)$	the isometry, diffeomorphism and homeomorphism groups
$\operatorname{Mod}(S)$, $\operatorname{Bir}(X)$	the mapping class group and the birational group
$(G,H)$, $G/H$	a Klein geometry and its homogeneous space