Group action

A group action (also called a group representation in the context of abstract algebra, though the term “representation” is often reserved for linear actions on vector spaces) is a formal way of describing symmetries of mathematical objects by means of a group. Formally, let $G$ be a group and $X$ a set. A (left) group action of $G$ on $X$ is a function

$$ \cdot : G \times X \to X,\qquad (g,x) \mapsto g\cdot x, $$

satisfying the two axioms:

1. Identity: For the identity element $e$ of $G$, $e\cdot x = x$ for all $x\in X$.
2. Compatibility: For all $g,h\in G$ and $x\in X$, $(gh)\cdot x = g\cdot (h\cdot x)$.

When the function is defined on the right, i.e., $x\cdot g$, the corresponding axioms are $x\cdot e = x$ and $(x\cdot g)\cdot h = x\cdot (gh)$; right actions are equivalent to left actions via inversion in the group.

Basic Concepts

• Orbit: The orbit of an element $x\in X$ under the action of $G$ is $\operatorname{Orb}_G(x)={g\cdot x \mid g\in G}$. Orbits partition $X$ into equivalence classes.
• Stabilizer (or isotropy group): The stabilizer of $x$ is $\operatorname{Stab}_G(x)={g\in G \mid g\cdot x = x}$. It is a subgroup of $G$.
• Transitive action: An action is transitive if there is a single orbit, i.e., for any $x,y\in X$ there exists $g\in G$ with $g\cdot x = y$.
• Free action: An action is free if $\operatorname{Stab}_G(x)={e}$ for all $x\in X$; equivalently, $g\cdot x = x$ implies $g=e$.
• Faithful (effective) action: The homomorphism $\phi:G\to \operatorname{Sym}(X)$ defined by $\phi(g)(x)=g\cdot x$ is injective. In other words, only the identity of $G$ acts as the identity permutation on $X$.

Algebraic Formulation

A group action can be equivalently described by a group homomorphism

$$ \phi : G \longrightarrow \operatorname{Sym}(X), $$

where $\operatorname{Sym}(X)$ denotes the symmetric group of all bijections of $X$. The image $\phi(G)$ is the permutation group induced by the action. Conversely, any homomorphism $\phi$ determines a unique action via $g\cdot x = \phi(g)(x)$.

When $X$ carries additional structure (e.g., a topological space, a manifold, a vector space, or an algebraic variety), one often requires the action to preserve that structure: continuous actions, smooth actions, linear actions, etc.

Examples

1. Permutation actions: The symmetric group $S_n$ acts on the set ${1,\dots,n}$ by permutation: $\sigma\cdot i = \sigma(i)$.
2. Matrix actions on vectors: The general linear group $GL_n(\mathbb{R})$ acts on $\mathbb{R}^n$ by matrix multiplication: $A\cdot v = Av$.
3. Rotations of the plane: The group $SO(2)$ of planar rotations acts on $\mathbb{R}^2$ via the usual rotation of vectors.
4. Conjugation action: Any group $G$ acts on itself by conjugation: $g\cdot x = gxg^{-1}$. The orbits are the conjugacy classes; stabilizers are the centralizers.
5. Action on coset spaces: For a subgroup $H\le G$, the left action of $G$ on the set of left cosets $G/H$ given by $g\cdot (aH) = (ga)H$ is transitive, and the stabilizer of the coset $H$ is precisely $H$.

Key Theorems

• Orbit–Stabilizer Theorem: For a finite group $G$ acting on a finite set $X$, $|G| = |\operatorname{Orb}_G(x)| \cdot |\operatorname{Stab}_G(x)|$ for any $x\in X$. This relates the size of the group to orbit and stabilizer sizes.
• Burnside’s Lemma (Cauchy–Frobenius Lemma): The number of orbits of a finite group $G$ acting on a finite set $X$ equals $\frac{1}{|G|}\sum_{g\in G} |\operatorname{Fix}(g)|$, where $\operatorname{Fix}(g)={x\in X\mid g\cdot x = x}$.
• Sylow’s Theorem (via group actions): Many proofs of Sylow’s theorems employ the action of a group on subsets of its elements or on coset spaces.

Applications

• Geometry and topology: Group actions describe symmetries of geometric objects, leading to concepts such as quotient spaces, orbifolds, and covering spaces.
• Representation theory: Linear actions of groups on vector spaces are studied as group representations, a central subject in algebra and physics.
• Combinatorics: Actions facilitate counting arguments (e.g., Polya enumeration) and the study of combinatorial structures up to symmetry.
• Dynamical systems: Continuous actions of groups (e.g., $\mathbb{R}$ or $\mathbb{Z}$) on spaces model time evolution.
• Physics: Symmetry groups act on configuration spaces, phase spaces, or fields; Noether’s theorem links continuous group actions to conservation laws.

Variants

• Right actions: Defined analogously with the compatibility condition $(x\cdot g)\cdot h = x\cdot (gh)$.
• Partial actions: The action map is defined only on a subset of $G\times X$; useful in operator algebras and dynamical systems.
• Groupoid actions: Generalize group actions by allowing the acting “group” to have multiple objects; relevant in foliation theory and noncommutative geometry.

References

• D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004.
• J. Rotman, An Introduction to the Theory of Groups, 4th ed., Springer, 1995.
• M. Artin, Algebra, 2nd ed., Pearson, 2011.
• H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Springer, 1980.