List of Group Actions

Introduction

This article lists the actions by which the corpus realises an abstract group as transformations of a set, an order, a vector space, an algebra or a field. For each action the list records the orbit, the stabiliser and the representation the action furnishes — the permutation representation, the defining representation, the regular representation, the Galois action or the action on a lattice of subgroups — together with the article that introduces it.

Every entry points to the article that introduces the action. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being an action that fails faithfulness or transitivity where its neighbour has it, with the failure named and the article that records it.

Actions on Sets

An action of $G$ on a set $X$ is a homomorphism $\rho : G \to \operatorname{Sym}(X)$; the orbit of a point is its equivalence class under the action, the stabiliser is the subgroup fixing it, and the orbit–stabiliser theorem relates the two to the order of the group. The action is faithful when the kernel of $\rho$ is trivial; the left regular action shows every group embeds in a symmetric group, which is Cayley's theorem.

Action	Orbit, stabiliser and representation	Introduced in
Left regular action of $G$ on itself	one orbit $G$; stabiliser trivial; the regular permutation representation, giving Cayley's embedding $G \hookrightarrow \operatorname{Sym}(G)$	Transformation Groups
Conjugation action of $G$ on itself	orbits the conjugacy classes; stabiliser $C_G(g)$, the centraliser; the permutation representation on conjugacy classes	Group Actions and Structure
Action of $G$ on the cosets $G/H$	one orbit; stabiliser a conjugate of $H$; a transitive $G$-set	Transformation Groups
Action of $G$ on its subgroups by conjugation	orbits the conjugacy classes of subgroups; stabiliser the normaliser $N_G(H)$	Group Actions and Structure
Action of $G$ on the power set of a $G$-set	orbits the sets of $k$-subsets; the induced permutation representation	Transformation Groups
Action of $G$ on itself by left multiplication, a monoid action	the monoid of endomorphisms; $\operatorname{Aut}(X) = \operatorname{End}(X)^\times$ for a structure $X$	Transformation Groups
Permutation representation	the homomorphism $\rho : G \to \operatorname{Sym}(X)$; faithful exactly when $\ker\rho = 1$	Transformation Groups
Non-example: the trivial action of a nontrivial group	fails faithfulness: the kernel is the whole group	Transformation Groups
Non-example: the action of $S_n$ on $\{1,\dots,n\}$ for $n \geq 3$	fails to be regular: the stabiliser of a point has order $(n-1)!$, not $1$	Transformation Groups

Actions on Lattices of Subgroups

The action-theoretic proofs of finite group theory use carefully chosen sets: the Sylow theorems apply the orbit–stabiliser theorem to the conjugation action on the Sylow subgroups, and the class equation counts the orbits of the conjugation action on the group itself. These actions act on a lattice of subgroups, ordered by inclusion, which is where the corpus meets the action of a group on an order.

Action	Orbit, stabiliser and representation	Introduced in
Conjugation action on $\operatorname{Syl}_p(G)$	the number of orbits $n_p$ divides the index; stabiliser the normaliser of a Sylow subgroup	Group Actions and Structure
Action of $G$ on the cosets of a Sylow subgroup	the representation $G \to S_{n_p}$ whose kernel controls the normal Sylow subgroup	Group Actions and Structure
Class equation	the orbit decomposition of conjugation; the fixed points are the centre $Z(G)$	Group Actions and Structure
Action of $G$ on its subgroups, ordered by inclusion	the lattice of subgroups is a $G$-poset; the orbits are the conjugacy classes of subgroups	Group Actions and Structure
Double-coset action of $H \times K$ on $G$	orbits the double cosets $HaK$; stabilisers the intersections $H \cap aKa^{-1}$	Group Actions and Structure
Burnside's lemma	the number of orbits as the average of the fixed-point counts	Group Actions and Structure
Non-example: the conjugation action of an abelian group	fails to be interesting only: every orbit is a point, since $C_G(g) = G$	Group Actions and Structure

Actions on Vector Spaces and Modules: the Representations

An action of $G$ on a vector space by linear maps is a representation of $G$, equivalently a module over the group algebra $F[G]$. The corpus meets the permutation representation of a group action, the defining representation of $GL(V)$, the regular representation of $G$ on $F[G]$, the dual and tensor representations, the symmetric and exterior powers, and the adjoint representation of a Lie algebra.

Action	Orbit, stabiliser and representation	Introduced in
Linear action of $GL(V)$ on $V$	the defining representation; the orbit of a vector is its nonzero scalar multiples, and the stabiliser the stabiliser of a line	The General Linear Group
Action of $GL(V)$ on the lines and Grassmannians	the orbits are the Grassmannians; the stabilisers the parabolic subgroups	The General Linear Group
Regular representation of $G$ on $F[G]$	$G$ acts on the group algebra; the character is $\chi_{\mathrm{reg}}(e) = \|G\|$	Group Algebras
Permutation representation of a $G$-set	the linearisation of an action on a set; the character counts fixed points	Representations of Groups
Tensor, dual, symmetric and exterior powers	the representations $V \otimes W$, $V^*$, $\operatorname{Sym}^n V$ and $\Lambda^n V$	Representations of Groups
Adjoint action of a Lie algebra on itself	orbits the coadjoint orbits; the representation $\operatorname{ad} : \mathrm{G} \to \mathrm{GL}(\mathrm{G})$	Lie Algebras
Weight decomposition	the action of a Cartan subalgebra; the weights and the highest weight	Representations of Lie Algebras
Schur–Weyl action of $S_n \times GL(V)$ on $V^{\otimes n}$	orbits and stabilisers described by Young diagrams; the decomposition into Schur functors	Schur–Weyl Duality
Non-example: a non-faithful representation	fails faithfulness: the kernel is a nontrivial normal subgroup	Representations of Groups
Non-example: a reducible representation in characteristic dividing $\|G\|$	fails complete reducibility: Maschke's hypothesis is defeated	Representations of Groups

Actions on Algebras and Fields

A group acts on an algebra by automorphisms, and the subalgebra it fixes is the ring of invariants; a group acts on a field extension as its Galois group, and the fixed field is the subject of the fundamental theorem. The action of a group on an algebra also builds new algebras, the crossed products, and the infinitesimal version of an automorphism is a derivation.

Action	Orbit, stabiliser and representation	Introduced in
Action of $\operatorname{Aut}(R)$ on a ring $R$	the fixed subring and the invariant centre; the transformation group of the ring	Ring and Field Automorphisms
Inner action $x \mapsto uxu^{-1}$ by a unit	the orbit is the conjugacy class; the action is by an inner automorphism	Ring and Field Automorphisms
Galois action of $\operatorname{Gal}(L/K)$ on $L$	the fixed field is $K$; the orbit of an element has size the degree of its minimal polynomial	Galois Theory
Action of $G$ on a polynomial ring by graded automorphisms	the invariant ring $R^G$; the Molien series computes the graded dimensions	Invariant Theory
Crossed product $A \# G$	the algebra built from a group action, with the action as twisted multiplication	Crossed Products
Action of $G$ on an algebra by automorphisms	the orbits of the automorphism group; the fixed subalgebra	Automorphisms and Derivations of Algebras
Derivation as the infinitesimal action	a derivation $\delta$ with the Leibniz rule; the inner derivations $\operatorname{ad}_a$	Automorphisms and Derivations of Algebras
Frobenius action on a finite field	the automorphism $x \mapsto x^p$, of order the degree over $\mathbb{F}_p$	Ring and Field Automorphisms
Non-example: an infinite Galois group of a finite extension	fails the finite-orbit description: the action is trivial on a finite Galois extension	Galois Theory
Non-example: an action of $\operatorname{Aut}(R)$ that fails to preserve the centre	does not occur: an automorphism preserves the centre, so the centre is a stabilised subring	Ring and Field Automorphisms

Actions on Spaces

Once a distance is available, a group acts on a topological space by homeomorphisms or by isometries, and the orbit–stabiliser theorem acquires the orbit map as a quotient map. The corpus treats these actions in Part II and Part III, and the algebraic layer records only the algebraic shadow of them.

Action	Orbit, stabiliser and representation	Introduced in
Action of a topological group on a space	orbits are the homogeneous spaces; the orbit map is continuous	Topological Groups (Part II)
Action of a discrete group on a space by deck transformations	the orbits are the fibres of a covering; the stabilisers are the deck group	The Fundamental Group and Covering Spaces (Part II)
Isometric action on a metric space	orbits are the equidistant sets; stabilisers are the isometry subgroups	Metric, Uniform and Complete Spaces (Part II)
Linear action of a Lie group on a manifold	the orbits are the homogeneous spaces $G/H$; the representation of the Lie algebra	Lie Groups (Part II)
Non-example: an action by homeomorphisms that is not by isometries	fails to preserve the distance: the orbit map is continuous but not Lipschitz	Metric, Uniform and Complete Spaces (Part II)

The Three Properties of an Action

Every action in the list is separated by three properties — whether it is faithful, whether it is transitive, and whether it is regular — and the table reads them side by side so that the same group appears with different properties on different sets.

Action	Faithful?	Transitive?	Introduced in
Left regular action of $G$ on itself	yes: the kernel is trivial	yes: one orbit	Transformation Groups
Action of $G$ on the cosets $G/H$	exactly when $H$ contains no nontrivial normal subgroup of $G$	yes, by construction	Transformation Groups
Conjugation action of $G$ on itself	exactly when the centre is trivial	exactly when $G$ is abelian	Group Actions and Structure
Trivial action of a nontrivial group	no: the kernel is the whole group	yes: one orbit	Transformation Groups
Action of $S_n$ on $\{1,\dots,n\}$, $n \geq 3$	yes	yes	Transformation Groups

Summary

The list gathers the group actions of the corpus. The actions on sets are the left regular action and Cayley's embedding, the conjugation action with the conjugacy classes and centralisers, the action on cosets, on subgroups, on power sets and the permutation representation. The actions on a lattice of subgroups are the conjugation action on the Sylow subgroups, the coset action on a Sylow subgroup, the class equation, the double-coset action and Burnside's lemma. The actions on vector spaces are the defining representation of $GL(V)$, its action on lines and Grassmannians, the regular representation on the group algebra, the permutation representation, the tensor, dual, symmetric and exterior power representations, the adjoint representation of a Lie algebra, the weight decomposition and the Schur–Weyl action on tensor powers. The actions on algebras and fields are the automorphism action on a ring, the inner action, the Galois action, the graded action on a polynomial ring with its invariant ring, the crossed product, the derivation as infinitesimal action and the Frobenius. The actions on spaces are the topological, covering, isometric and Lie-group actions of Parts II and III. The non-examples — the trivial action, the non-regular action on a finite set, the conjugation action of an abelian group, a non-faithful representation, a reducible representation in the modular case, an action failing to preserve the centre, and an action by homeomorphisms that is not isometric — each name the property that fails.

Summary of Notation

The article denotes its actions by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$\operatorname{Sym}(X)$, $S_n$	symmetric group on a set, on $n$ letters
$\rho : G \to \operatorname{Sym}(X)$	permutation representation of an action
$\operatorname{Orb}(x)$, $\operatorname{Stab}(x)$	orbit and stabiliser
$C_G(g)$, $Z(G)$, $N_G(H)$	centraliser, centre, normaliser
$\operatorname{Syl}_p(G)$, $n_p$	Sylow $p$-subgroups and their number
$F[G]$, $\chi_V$, $\operatorname{Rep}_F(G)$	group algebra, character, representation category
$V \otimes W$, $V^*$, $\operatorname{Sym}^n V$, $\Lambda^n V$	tensor, dual, symmetric and exterior power representations
$\operatorname{ad}$	adjoint representation of a Lie algebra
$R^G$	ring of invariants
$\operatorname{Gal}(L/K)$	Galois group of a field extension
$\operatorname{Aut}(R)$, $\operatorname{Inn}(G)$	automorphism group; inner automorphisms
$\mathbb{F}_p$, $\mathbb{F}_q$	finite fields