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Group action

In mathematics, a group action is a fundamental concept in abstract algebra where a group G operates on a set X through transformations that preserve the group's structure, effectively describing symmetries or permutations induced by the group elements.^[1] Formally, a group action is a function \phi: G \times X \to X, often denoted (g, x) \mapsto g \cdot x, satisfying two axioms: the identity element e \in G acts trivially, so e \cdot x = x for all x \in X, and the action is compatible with group multiplication, so g \cdot (h \cdot x) = (gh) \cdot x for all g, h \in G and x \in X.^[2] This structure turns X into a G-set, allowing the group to "act" on it in a consistent manner.^[3] Group actions generalize the notion of symmetry groups, such as the dihedral group acting on the vertices of a regular polygon by rotations and reflections, and are essential for analyzing how groups interact with geometric, algebraic, or combinatorial objects.^[4] Central to the theory are orbits and stabilizers: the orbit of x \in X is the set \{ g \cdot x \mid g \in G \}, representing all points reachable from x via the action, while the stabilizer of x is the subgroup \{ g \in G \mid g \cdot x = x \}, consisting of elements that fix x.^[1] The orbit-stabilizer theorem establishes a key relationship: if G is finite, the size of the orbit equals the index of the stabilizer in G, providing a tool to compute cardinalities and classify actions as transitive (single orbit), free (trivial stabilizers), or faithful (injective action map).^[2] Beyond basic properties, group actions enable profound applications across mathematics. In Galois theory, the Galois group acts on the roots of a polynomial, with orbits corresponding to irreducible factors, linking algebra to field extensions.^[1] In combinatorics, Burnside's lemma uses fixed points of group elements to count distinct objects under symmetry, such as necklace colorings or molecular configurations.^[5] Actions also extend to topological and geometric settings, where continuous group actions model symmetries in manifolds or spaces, influencing representation theory, cohomology, and physics via symmetry groups in quantum mechanics and particle physics.^[4] These concepts, first systematically explored in the context of permutation groups and Galois correspondences in the 19th century, remain indispensable for proving theorems like Sylow's in finite group theory and understanding modular forms.^[2]

Definitions

Left actions

A left action of a group G on a set X is a map G \times X \to X, denoted (g, x) \mapsto g \cdot x, satisfying the identity condition e \cdot x = x for all x \in X, where e is the identity element of G, and the compatibility condition (gh) \cdot x = g \cdot (h \cdot x) for all g, h \in G and x \in X.^[6] These axioms ensure that the action respects the group structure, allowing elements of G to "transform" points in X in a consistent manner that mirrors group multiplication.^[2] The notation for such an action is commonly expressed as G acting on X, with g \cdot x often simplified to gx.^[2] This shorthand facilitates concise descriptions in algebraic contexts, emphasizing the operational nature of the action.^[6] The structured pair (X, \cdot), where the action is specified, is termed a G-set, encapsulating both the underlying set and the group operation on it.^[6] The concept of group actions has roots in 19th-century permutation group theory, with the modern abstract definition formalized in early 20th-century developments in group theory.^[1]

Right actions

A right action of a group G on a set X is a function X \times G \to X, written (x, g) \mapsto x \cdot g, that satisfies two axioms: the identity element e \in G acts trivially, so x \cdot e = x for all x \in X, and the action respects the group operation from the right, so (x \cdot g) \cdot h = x \cdot (gh) for all x \in X and g, h \in G.^[2] This contrasts with the standard left action, where the group operation is applied from the left: g \cdot (h \cdot x) = (gh) \cdot x.^[3] The notation for right actions often uses exponentiation x^g to denote x \cdot g, or simply juxtaposition xg, emphasizing the rightward application.^[2] Right actions arise naturally in contexts like permutation representations where the order of group elements matters for composition, such as when viewing permutations as acting on the right in certain symmetry studies.^[5] However, right actions are equivalent to left actions up to inversion of group elements. Specifically, given a right action x \cdot g, one defines a corresponding left action by g \cdot x = x \cdot g^{-1}; this map satisfies the left action axioms because inversion turns the right compatibility into left compatibility: g \cdot (h \cdot x) = (x \cdot h^{-1}) \cdot g^{-1} = x \cdot (h^{-1} g^{-1}) = x \cdot (gh)^{-1} = (gh) \cdot x. The converse holds similarly, establishing a bijection between right G-actions on X and left G-actions on X.^[7] When the group G is abelian, left and right actions coincide in a stronger sense because the inversion map g \mapsto g^{-1} is itself a group homomorphism: (gh)^{-1} = g^{-1} h^{-1} = h^{-1} g^{-1}. This makes the equivalence canonical and preserves the group structure directly.^[8]

Fundamental Properties

A group action of a group G on a set X is called transitive if for every pair of elements x, y \in X, there exists an element g \in G such that g \cdot x = y.^[9] Equivalently, the action is transitive if and only if X forms a single orbit under the action of G.^[9] A stronger notion is that of a doubly transitive action, where for any two ordered pairs of distinct elements (x_1, x_2), (y_1, y_2) \in X \times X with x_1 \neq x_2 and y_1 \neq y_2, there exists g \in G such that g \cdot x_1 = y_1 and g \cdot x_2 = y_2.^[10] Every doubly transitive action is transitive, but the converse does not hold in general.^[11] An action is termed faithful if the associated homomorphism \phi: G \to \mathrm{Sym}(X) from G to the symmetric group on X is injective, meaning the kernel is trivial and only the identity element of G fixes every point in X.^[12] In other words, if g \cdot x = x for all x \in X implies g is the identity.^[12] Finally, an action is regular if it is both faithful and transitive, and in such cases, the cardinalities satisfy |G| = |X|.^[13] Equivalently, for any x, y \in X, there is exactly one g \in G such that g \cdot x = y.^[13] Regular actions provide a canonical way to realize G as a permutation group on a set of the same size.^[13]

Primitivity

In group theory, a subset B \subseteq X of the set X on which a group G acts is called a block if $1 < |B| < |X| and for every g \in G, either gB = B or gB \cap B = \emptyset.^[14] Such blocks correspond to the parts of nontrivial partitions of X that are preserved by the action of G. A group action of G on a set X is primitive if it is transitive and admits no nontrivial blocks.^[15] Equivalently, the only G-invariant partitions of X are the trivial ones consisting of singletons or the full set \{X\}.^[16] Primitivity thus captures a notion of "maximal transitivity" in the sense that the action cannot be refined into a coarser transitive action on a system of blocks without losing transitivity on X. A key characterization of primitive actions is that a transitive action of G on X is primitive if and only if the stabilizer G_x of any point x \in X is a maximal subgroup of G.^[17] This equivalence highlights the structural rigidity of primitive actions: the point stabilizers leave no room for intermediate subgroups that could induce nontrivial block systems. In the context of permutation groups, this means that primitive subgroups of the symmetric group S_n arise precisely as the images of transitive actions where stabilizers are maximal. Primitive actions are particularly significant for simple groups, as their faithful primitive representations embed the group as a primitive permutation subgroup of some symmetric group S_n, providing insights into the embedding problem and the classification of finite simple groups via permutation representations.^[18]

Orbits and Stabilizers

Orbits and invariant subsets

The orbit of an element x \in X under a group action of G on X is the set

\Orb_G(x) = \{ g \cdot x \mid g \in G \},

consisting of all elements of X that can be obtained by acting on x with elements of G. This set is the equivalence class of x under the relation \sim defined by x \sim y if and only if there exists g \in G such that y = g \cdot x; the relation \sim is an equivalence relation on X.^[19]^[2] The equivalence relation induced by the group action partitions the set X into disjoint orbits, decomposing X as a disjoint union of these orbits. Each orbit is stable under the action, and the restriction of the action to an orbit yields a transitive action on that orbit.^[20] A subset Y \subseteq X is invariant under the group action if g \cdot Y = Y for all g \in G. Equivalently, every invariant subset is a union of orbits, and conversely, every union of orbits is invariant. Thus, the collection of invariant subsets corresponds precisely to the subsets of X that are saturated with respect to the partition into orbits.^[21]^[22]

Stabilizers and fixed points

In group theory, given a group G acting on a set X, the stabilizer of an element x \in X, denoted \operatorname{Stab}_G(x) or simply \operatorname{Stab}(x), is the subgroup consisting of all elements of G that fix x under the action.^[2] Specifically,

\operatorname{Stab}_G(x) = \{ g \in G \mid g \cdot x = x \}.

This set forms a subgroup of G because it is nonempty (containing the identity element), closed under the group operation (if g \cdot x = x and h \cdot x = x, then (gh) \cdot x = g \cdot (h \cdot x) = g \cdot x = x), and closed under inverses (if g \cdot x = x, then g^{-1} \cdot x = x by applying g^{-1} to both sides of the equation). Different elements of X may have distinct stabilizers, and the stabilizer captures the "symmetry" preserving a particular point.^[2] For a fixed group element g \in G, the fixed points of g are the elements of X that remain unchanged when acted upon by g. Formally, the fixed-point set of g, denoted \operatorname{Fix}_G(g) or X^g, is defined as

\operatorname{Fix}_G(g) = \{ x \in X \mid g \cdot x = x \}.

This set measures the extent to which g acts trivially on X, and it always includes the points stabilized by g alone.^[2] The identity element e \in G has \operatorname{Fix}_G(e) = X, as it fixes every point by definition. An action of G on X is termed fixed-point-free if the only element of G with nonempty fixed-point set beyond the trivial case is the identity, meaning \operatorname{Fix}_G(g) = \emptyset for all g \in G \setminus \{e\}.^[2] Equivalently, such an action is called free, where every non-identity element moves all points in X, implying that stabilizers are trivial (\operatorname{Stab}_G(x) = \{e\} for all x \in X). Free actions are fundamental in studying coverings and quotients, as they ensure the action is "as injective as possible" on points.^[2] When the set X is itself a group (say H) and G acts on H by conjugation, the stabilizer of an element h \in H coincides with the centralizer of h in G, defined as \{ g \in G \mid g h g^{-1} = h \}, or equivalently \{ g \in G \mid g h = h g \}. This specializes the general stabilizer concept to actions preserving the group structure, highlighting commutativity relations within G.^[2]

Orbit-stabilizer theorem

The orbit-stabilizer theorem establishes a fundamental relationship between the orbit of an element under a group action and the stabilizer of that element. For a group G acting on a set X, and for any x \in X, the cardinality of the orbit \operatorname{Orb}_G(x) equals the index of the stabilizer \operatorname{Stab}_G(x) in G:

|\operatorname{Orb}_G(x)| = [G : \operatorname{Stab}_G(x)].

If G is finite, this simplifies to |\operatorname{Orb}_G(x)| = |G| / |\operatorname{Stab}_G(x)|.^[2]^[23] The proof proceeds by constructing a bijection between the set of left cosets G / \operatorname{Stab}_G(x) and \operatorname{Orb}_G(x). Define the map \phi: G / \operatorname{Stab}_G(x) \to \operatorname{Orb}_G(x) by \phi(g \operatorname{Stab}_G(x)) = g \cdot x. This is well-defined because if g \operatorname{Stab}_G(x) = g' \operatorname{Stab}_G(x), then g'^{-1} g \in \operatorname{Stab}_G(x), so g' \cdot x = g \cdot x. It is surjective since every element in the orbit is g \cdot x for some g \in G. It is injective because if \phi(g \operatorname{Stab}_G(x)) = \phi(g' \operatorname{Stab}_G(x)), then g \cdot x = g' \cdot x, implying g^{-1} g' \in \operatorname{Stab}_G(x) and thus g \operatorname{Stab}_G(x) = g' \operatorname{Stab}_G(x). For a right action, the proof uses right cosets analogously.^[23] When G is finite and the action is transitive (i.e., X is a single orbit), the theorem implies that |X| divides |G|, since |\operatorname{Stab}_G(x)| divides |G| by Lagrange's theorem. In a regular action, where the action is both transitive and free (stabilizers are trivial), it follows that |X| = |G|. For infinite groups, the bijection ensures that the cardinality of the orbit equals the cardinality of the index [G : \operatorname{Stab}_G(x)], providing a cardinal arithmetic analogue.^[2]^[23]

Burnside's lemma

Burnside's lemma, also known as the Cauchy–Frobenius lemma, is a fundamental result in group theory that enumerates the number of orbits in a group action on a set. For a finite group G acting on a finite set X, the lemma states that the number of orbits |X / G| is equal to the average number of points fixed by each group element:

|X / G| = \frac{1}{|G|} \sum_{g \in G} |\mathrm{Fix}(g)|,

where \mathrm{Fix}(g) = \{ x \in X \mid g \cdot x = x \} denotes the fixed points of g. This formula aggregates the fixed-point data over the entire group to yield a global count of distinct orbits under the action. The proof proceeds by double counting the set of pairs (g, x) \in G \times X such that g \cdot x = x. The size of this set is \sum_{g \in G} |\mathrm{Fix}(g)|. Alternatively, fixing x \in X first, the number of such g is |\mathrm{Stab}(x)|, so the total is \sum_{x \in X} |\mathrm{Stab}(x)|. By the orbit-stabilizer theorem, |\mathrm{Stab}(x)| = |G| / |\mathrm{Orbit}(x)|, yielding \sum_{x \in X} |G| / |\mathrm{Orbit}(x)| = |G| \sum_{x \in X} 1 / |\mathrm{Orbit}(x)|. The inner sum equals the number of orbits, since each orbit contributes exactly once regardless of its size. Thus, |X / G| = \frac{1}{|G|} \sum_{g \in G} |\mathrm{Fix}(g)|. For infinite groups, a version of Burnside's lemma holds when G is a compact topological group equipped with a normalized Haar measure \mu (satisfying \mu(G) = 1). If G acts continuously on a discrete set X with finitely many orbits, the number of orbits is \int_G |\mathrm{Fix}(g)| \, d\mu(g). This integral generalizes the finite average, leveraging the invariance of the Haar measure under group translations. A significant combinatorial application of Burnside's lemma is Pólya's enumeration theorem, which originated as a method to count the number of distinct colorings of a structure up to group symmetries by incorporating the cycle structures of group elements into a generating function known as the cycle index.^[24]

Examples and Applications

Classical examples

One of the most fundamental examples of a group action is the natural action of the symmetric group S_n on the finite set X = \{1, 2, \dots, n\}, where each permutation \sigma \in S_n acts by \sigma \cdot i = \sigma(i) for i \in X.^[2] This action is faithful, meaning the kernel of the corresponding homomorphism S_n \to \mathrm{Sym}(X) is trivial, as distinct permutations move elements differently.^[6] It is also transitive, since any two elements in X can be mapped to each other by some permutation, and primitive for n \geq 2, as the only blocks are trivial singletons or the full set.^[2] Another classical example involves the cyclic group C_n = \langle r \rangle of order n, acting regularly on the set of n-th roots of unity in the complex numbers, denoted \mu_n = \{ e^{2\pi i k / n} \mid k = 0, 1, \dots, n-1 \}, via multiplication: r^k \cdot \zeta = r^k \zeta for \zeta \in \mu_n.^[25] Here, \mu_n itself forms a cyclic group under multiplication, isomorphic to C_n, and the action is regular because the stabilizer of any root is trivial and the orbit of any root is the entire set. The dihedral group D_n of order $2n, consisting of the symmetries of a regular n-gon (rotations and reflections), acts on the set of n vertices of the polygon by rigidly mapping the figure to itself.^[26] This action is transitive, as any vertex can be mapped to any other via a suitable rotation or reflection.^[26] Furthermore, it is primitive if and only if n is prime, since non-prime n allows nontrivial blocks corresponding to divisors of n.^[27] A simple yet illustrative example is the trivial action of any group G on an arbitrary nonempty set X, defined by g \cdot x = x for all g \in G and x \in X. In this case, every element of X is a fixed point, so each singleton \{x\} is both an orbit and a fixed set, and the action has trivial stabilizers everywhere.^[2] Finally, every group G acts on itself by conjugation, where g \cdot h = g h g^{-1} for g, h \in G.^[28] The orbits of this action are precisely the conjugacy classes of G, which partition G into subsets of elements sharing the same cycle type (in the symmetric group case) or equivalent under inner automorphisms more generally.^[28] The stabilizer of h is the centralizer C_G(h), and this action is useful for studying the structure of G via its class equation.^[28]

Combinatorial applications

Group actions find prominent applications in combinatorics, particularly in enumerating distinct objects up to symmetry through the use of Burnside's lemma, which counts the orbits of the action.^[29] This approach is essential for problems where symmetries, such as rotations or reflections, identify equivalent configurations, allowing the computation of inequivalent structures by averaging the fixed points over the group elements.^[30] A classic example is the counting of necklaces, where the cyclic group C_n acts on the set of colorings of n beads with a fixed number of colors. Here, two colorings are considered the same if one can be obtained from the other by rotation, and Burnside's lemma determines the number of distinct necklaces by summing the fixed colorings for each group element and dividing by the group order.^[31] For instance, with c colors, the formula yields the number of rotationally distinct arrangements, providing a practical tool for such symmetric enumerations.^[32] In graph theory, the automorphism group \operatorname{Aut}(G) of a graph G acts on the set of proper k-colorings, where the orbits correspond to colorings that are inequivalent under the graph's symmetries. Applying Burnside's lemma counts these orbits, yielding the number of distinct colorings up to automorphism, which is useful for classifying symmetric graph structures.^[33] Burnside's lemma also appears in the enumeration of orbits within combinatorial species, where group actions on labeled structures help count unlabeled species by considering symmetries in generating functions. Historically, Burnside introduced the lemma in his 1897 book Theory of Groups of Finite Order, applying it to combinatorial problems in finite group theory.^[34]

Morphisms of Actions

Homomorphisms between G-sets

A homomorphism between two G-sets X and Y, also known as a G-equivariant map, is a function \phi: X \to Y satisfying \phi(g \cdot x) = g \cdot \phi(x) for all g \in G and x \in X.^[6] This condition ensures that \phi intertwines the actions of G on X and Y, preserving the group action structure.^[6] The collection of all G-sets, together with these G-homomorphisms as morphisms, forms a category denoted \mathbf{Set}^G.^[35] Any group action of G on a set X induces a permutation representation, which is a group homomorphism \rho: G \to \mathrm{Sym}(X) defined by \rho(g)(x) = g \cdot x for all g \in G and x \in X, where \mathrm{Sym}(X) is the symmetric group on X.^[36] This homomorphism embeds the action into the category of permutation groups, allowing the study of actions via their images in \mathrm{Sym}(X).^[37] The kernel of a group action on X is the set \ker(\rho) = \{ g \in G \mid g \cdot x = x \ \forall x \in X \}, which coincides with the kernel of the induced homomorphism \rho: G \to \mathrm{Sym}(X).^[2] As the kernel of a group homomorphism, \ker(\rho) is a normal subgroup of G.^[38] An action is faithful if and only if this kernel is trivial.^[38] By the first isomorphism theorem for groups, the image \mathrm{im}(\rho) is a subgroup of \mathrm{Sym}(X) isomorphic to the quotient group G / \ker(\rho).^[2] This isomorphism identifies the effective action of G modulo its kernel with a concrete permutation group acting on X.^[2] G-equivariant maps preserve orbits, mapping the orbit of x \in X into the orbit of \phi(x) \in Y.^[6]

Isomorphisms and conjugacy

A G-isomorphism between two G-sets (X, \cdot) and (Y, \circ) is a bijective map f: X \to Y that is a G-homomorphism, meaning f(g \cdot x) = g \circ f(x) for all g \in [G](/page/G) and x \in X, with the additional property that its inverse f^{-1}: Y \to X is also a G-homomorphism. Since f is bijective, the equivariance of the inverse follows automatically from the group action axioms, ensuring that isomorphisms preserve the entire structure of the actions.^[2]^[39] Two group actions on sets X and Y are said to be equivalent if their corresponding G-sets are isomorphic, i.e., there exists a G-isomorphism between them. This equivalence relation classifies actions up to structural similarity, preserving key features such as the orbit-stabilizer theorem: isomorphic G-sets have corresponding orbits of equal cardinality and stabilizers that are isomorphic as subgroups of G. For instance, the left regular action of G on itself is equivalent to any transitive free action on a set of cardinality |G|.^[2]^[6] Conjugate actions provide another way to relate actions of the same group G on a fixed set X. Given an action \cdot : G \times X \to X and an automorphism \alpha \in \Aut(G), the conjugate action \cdot_\alpha is defined by

g \cdot_\alpha x = \alpha(g) \cdot x

for all g \in G and x \in X. This defines a new group action, as \alpha preserves the group operation, and it twists the original action by relabeling elements of G via \alpha. The conjugation action of G on itself, defined by g \cdot x = g x g^{-1} for g, x \in G, is a related example, yielding orbits as conjugacy classes. Cayley's theorem illustrates these concepts through the regular action: every group G embeds as a subgroup of the symmetric group \Sym(G) via the left regular action \lambda: G \to \Sym(G) defined by \lambda(g)(h) = gh for g, h \in G, yielding a faithful transitive action isomorphic to the standard permutation representation. This embedding is unique up to conjugacy in \Sym(G), meaning any two such regular embeddings differ by conjugation by some permutation in \Sym(G), reflecting the equivalence of regular actions under relabeling of the set.^[40]^[41]

Advanced Variants

Topological group actions

A topological group G acts continuously on a topological space X if the action map \mu: G \times X \to X, defined by \mu(g, x) = g \cdot x, is continuous with respect to the product topology on G \times X.^[42] This joint continuity ensures that the action respects the topological structures of both G and X, distinguishing it from merely set-theoretic actions by requiring that nearby group elements act in a controlled manner on points in X. In many cases, such as when G is a Lie group and X is locally compact, this implies that each fixed g \in G acts via a homeomorphism on X, though the converse requires additional assumptions like separate continuity.^[43] Topological transitivity extends the algebraic notion of transitivity to the continuous setting: an action is topologically transitive if, for every pair of nonempty open sets U, V \subseteq X, there exists g \in G such that g \cdot U \cap V \neq \emptyset, or equivalently, the orbit G \cdot U is dense in X.^[44] In compact metric spaces, this is equivalent to the existence of at least one dense orbit.^[45] Topological transitivity captures mixing or ergodic-like behavior in dynamical systems arising from group actions, where orbits densely fill the space without collapsing to finite sets as in the discrete case. A continuous action is proper if the map G \times X \to X \times X, (g, x) \mapsto (x, g \cdot x), is a proper map, meaning that the preimage of every compact subset of X \times X is compact in G \times X.^[46] Equivalently, for every compact K \subseteq X, the set \{(g, x) \in G \times X \mid x \in K, g \cdot x \in K\} is compact.^[47] Proper actions ensure well-behaved orbit spaces and stabilizers, often compact, which is crucial for quotient constructions like orbifolds; compact groups always act properly on Hausdorff spaces.^[48] A classic example is the continuous action of the special orthogonal group SO(3), the group of 3D rotations, on \mathbb{R}^3 \setminus \{0\} by matrix-vector multiplication: for A \in SO(3) and v \in \mathbb{R}^3 \setminus \{0\}, A \cdot v = Av. This action is continuous since matrix multiplication is a polynomial map, hence continuous, and proper because SO(3) is compact.^[49] The orbits are spheres centered at the origin, illustrating how properness prevents "accumulation at infinity." In modern dynamical systems, such topological group actions underpin the study of qualitative behaviors like chaos and recurrence on non-compact spaces, extending classical ergodic theory to broader group structures.

Linear representations

A linear group action on a vector space arises when a group G acts on a vector space V over a field k in a manner that respects the linear structure of V. Specifically, the action satisfies g \cdot (a v + b w) = a (g \cdot v) + b (g \cdot w) for all g \in G, scalars a, b \in k, and vectors v, w \in V.^[50] Such actions are equivalently described by group homomorphisms \rho: G \to \GL(V), where \GL(V) is the general linear group of invertible linear endomorphisms of V; here, g \cdot v = \rho(g) v.^[50] These homomorphisms are termed linear representations of G on V. Associated to any such representation is its character \chi: G \to k, defined by \chi(g) = \trace(\rho(g)), the trace of the linear map \rho(g).^[50] An invariant subspace under a linear representation \rho is a subspace W \subseteq V such that \rho(g) w \in W for all g \in G and w \in W. A representation is irreducible if the only invariant subspaces are \{0\} and V itself, meaning no nontrivial proper invariant subspaces exist.^[50] Irreducible representations form the building blocks of more general representations, as they cannot be decomposed further into simpler linear actions. For finite groups G, Maschke's theorem provides a key decomposition result: if the characteristic of k does not divide the order |G|, then every finite-dimensional representation of G over k is completely reducible, meaning it decomposes as a direct sum of irreducible representations.^[50] This holds in particular over fields of characteristic zero, such as the complex numbers, but also over finite fields when the characteristic avoids dividing |G|; in cases where the characteristic divides |G|, representations may fail to be completely reducible, as seen in modular representation theory over fields like \mathbb{F}_p for p-groups.^[50]

Actions on groupoids

A groupoid is defined as a small category in which every morphism is an isomorphism.^[51] This structure generalizes groups, which are groupoids with a single object, by allowing multiple objects connected by invertible arrows representing symmetries or equivalences between them.^[51] An action of a group G on a groupoid \mathcal{C} consists of actions of G on both the set of objects \mathrm{Ob}(\mathcal{C}) and the set of morphisms \mathrm{Mor}(\mathcal{C}), compatible with the category structure of \mathcal{C}. Specifically, for each g \in G, the map on morphisms g \cdot (-) : \mathrm{Mor}(\mathcal{C}) \to \mathrm{Mor}(\mathcal{C}) satisfies s(g \cdot f) = g \cdot s(f) and t(g \cdot f) = g \cdot t(f) for source s and target t, preserves composition via g \cdot (f \circ h) = (g \cdot f) \circ (g \cdot h), and maps identities to identities g \cdot \mathrm{id}_x = \mathrm{id}_{g \cdot x}.^[52] This compatibility ensures that each g induces an automorphism of the groupoid \mathcal{C}, and the map G \to \mathrm{Aut}(\mathcal{C}) is a group homomorphism.^[53] Ordinary group actions on sets recover this framework when \mathcal{C} is the discrete groupoid on the set, with only identity morphisms. In geometric contexts, actions on groupoids extend to pseudogroup actions, where a pseudogroup—a collection of local homeomorphisms satisfying group-like axioms—acts on a space by generating local transformations, unifying global group actions with infinitesimal symmetries like Lie algebra actions.^[54] Such structures are essential for studying foliations and orbifolds, where the pseudogroup encodes local equivalence relations.^[54] Modern developments in higher category theory, particularly Baez's work on groupoidification in the 2000s, reinterpret group actions on groupoids categorically to "categorify" linear algebra: vector spaces become groupoids, and linear maps become spans of groupoids, yielding insights into representation theory and quantum protocols via weak quotients and orbit-stabilizer relations.^[55] Morphisms of such actions, generalizing G-set homomorphisms, are functors intertwining the group actions.^[53]

Generalizations

Partial actions

In group theory, a partial action of a discrete group G on a set X generalizes the classical notion of a group action by allowing the action of certain group elements to be undefined on parts of the set. Specifically, it consists of a family of subsets \{D_g \subseteq X \mid g \in G\}, called domains, together with bijections \theta_g : D_{g^{-1}} \to D_g for each g \in G, satisfying the following conditions: D_e = X and \theta_e is the identity map on X, where e is the group identity; for all g, h \in G, the set equality D_{gh} = \theta_g(D_{h^{-1}}) holds, and the maps compose as \theta_{gh} = \theta_g \circ \theta_h whenever the composition is defined, i.e., on D_{h^{-1} g^{-1}}. The partial action is then denoted by g \cdot x = \theta_g(x) for x \in D_{g^{-1}}, with the action undefined otherwise. This structure ensures partial compatibility with the group operation, preserving bijectivity where defined.^[56] A set X equipped with such a partial action is termed a partial G-set. The domains D_g often exhibit additional structure, such as forming a graded family where intersections and images align with the group multiplication, enabling the partial G-set to behave like a subsystem of a full action. For instance, invariant subsets of X under the partial action can naturally serve as domains in extensions or restrictions of the structure. Partial G-sets generalize full G-sets, where D_g = X for all g, and every partial action embeds into a full action on a larger set via constructions like the universal enveloping action, which adjoins elements to make the action total while preserving the original partial dynamics.^[56] Partial actions find significant applications in ring theory, where they extend to actions on algebras, allowing groups to act partially on non-unital or semiprime rings. In this context, a partial action \alpha of G on a ring R assigns to each g \in G an ideal A_g \subseteq R (playing the role of D_g) and a ring isomorphism \alpha_g : A_{g^{-1}} \to A_g, with analogous compatibility conditions. This leads to the partial skew group ring R \rtimes^\alpha G, a generalization of the classical skew group ring that captures crossed product-like structures even when full actions are unavailable. The theory emerged in the late 1990s, with foundational work establishing conditions for globalization—embedding partial actions into full actions on larger rings—and associativity of the partial skew group ring, particularly for s-unital rings.^[57] These algebraic partial actions have facilitated developments in noncommutative Galois theory, where they describe extensions of rings under group symmetries that are not globally defined, and in the classification of simple Artinian rings via partial Galois correspondences. For semiprime rings, every partial action admits a weak enveloping action, ensuring the partial skew group ring is well-behaved and Morita equivalent to a full crossed product under certain regularity conditions. This framework, built on contributions from the 2000s onward, underscores partial actions' role in bridging set-theoretic dynamics with algebraic invariants.

Mackey functors

In equivariant homotopy theory and representation theory, Mackey functors provide a generalization of group actions by axiomatizing the induction, restriction, and conjugation operations that arise in the study of G-sets and representations for a finite group G over a commutative ring R. A Mackey functor M assigns to each subgroup H ≤ G an abelian group M(H), together with natural maps: restriction res^K_H : M(H) → M(K) for K ≤ H, induction ind^H_K : M(K) → M(H), and transfer (or conjugation) tr^g : M(H) → M(^g H) for g ∈ G, satisfying axioms such as res^G_H ∘ ind^G_K = ind^H_K ∘ res^K_H (projection formula), double coset decompositions (Mackey formula), and Frobenius reciprocity.^[58] Mackey functors generalize the representation theory of finite groups, where the category of R G-modules forms a Mackey functor via Hom and tensor operations. They form an abelian category Mack_R(G), with examples including the Burnside ring (counting orbits), cohomology groups H^n(G, U), and K-groups of group rings. This structure captures the dynamics of group actions on categories rather than sets, enabling computations in equivariant cohomology and stable homotopy.^[58] Mackey functors have applications in algebraic topology, where they classify equivariant spectra and compute homotopy groups, and in modular representation theory for decomposing induced modules via Mackey decomposition formulas. Historically, the concept was formalized in the 1970s–1980s building on George Mackey's work on induced representations, with key developments by Peter Webb and others in the 1990s for functorial approaches to group cohomology.^[59]^[58]

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