Group Action

Overview

A group action connects abstract multiplication with transformations of a concrete set. It can describe rotations of vertices, permutations of roots, or multiplication on cosets. Actions organize elements into orbits and measure their symmetries through stabilizers, making them both a structural tool and a counting method.

Prerequisites

Know groups, subgroups, and bijections. For a set $X$, its permutations form a group $\operatorname{Sym}(X)$ under composition. A group homomorphism preserves multiplication. These facts provide an equivalent formulation of the action axioms.

Definition and conventions

A left action of a group $G$ on a set $X$ is a function $G\times X\to X$ satisfying

for all $g,h\in G$ and $x\in X$. Each $g$ acts bijectively, with inverse transformation given by $g^{-1}$. Thus an action is equivalently a homomorphism $G\to\operatorname{Sym}(X)$.

The orbit of $x$ is $Gx=\{gx:g\in G\}$. Its stabilizer is $G_x=\{g:gx=x\}$. The action is faithful if only the identity fixes every point, and transitive if $X$ is nonempty and has one orbit. A right action has a different order convention; it must not be substituted silently into these formulas.

Intuition

The orbit asks where a point can go. The stabilizer asks which transformations leave it where it is. An action may hide some of the group: every element in the kernel of the permutation homomorphism is invisible on the entire set.

Examples and non-examples

The square's symmetry group has eight elements and acts transitively on its four vertices. A chosen vertex is fixed by the identity and the reflection in the diagonal through it. Its stabilizer therefore has two elements, and the orbit has $8/2=4$ points.

Any group acts trivially on a singleton. This action is transitive but, for a nontrivial group, not faithful. These conditions are independent.

A proposed action of the cyclic group of order two that assigns a three-cycle to its generator fails: the square of the assigned permutation is not the identity. Assigning a bijection to each element is insufficient without compatibility.

Important properties

Orbits partition $X$. The map $gG_x\mapsto gx$ gives a bijection $G/G_x\to Gx$. For finite $G$, it yields the orbit–stabilizer formula

To check well-definedness, two representatives differ by an element fixing $x$. Surjectivity follows from the orbit definition, and injectivity reverses that same calculation.

For a finite action, averaging the numbers of fixed points gives the number of orbits. This counts equivalent configurations without choosing representatives.

Connections and further reading

The conjugation action leads to conjugacy classes and the class equation. Linear actions are representations. See Milne, Chapter 4, Definitions and examples, Orbits, and Stabilizers.