# Group action

An action of a group \(G\) on a set \(X\) is a function \(\alpha : G \times X \to X\) (colon-to notation), which is often written \((g, x) \mapsto gx\) (mapsto notation), with \(\alpha\) omitted from the notation, such that

\(ex = x\) for all \(x \in X\), where \(e\) is the identity, and

\(g(hx) = (gh)x\) for all \(g, h \in G, x \in X\), where \(gh\) implicitly refers to the group operation in \(G\) (also omitted from the notation).

Equivalently, via currying, an action of \(G\) on \(X\) is a group homomorphism \(G \to \text{Aut}(X)\), where \(\text{Aut}(X)\) is the automorphism group of \(X\) (so for sets, the group of all bijections \(X \to X\), but phrasing the definition this way makes it natural to generalize to other categories). It’s a good exercise to verify this; Arbital has a proof.

Group actions are used to make precise the notion of “symmetry” in mathematics.

# Examples

Let \(X = \mathbb{R}^2\) be the Euclidean plane. There’s a group acting on \(\mathbb{R}^2\) called the Euclidean group \(ISO(2)\) which consists of all functions \(f : \mathbb{R}^2 \to \mathbb{R}^2\) preserving distances between two points (or equivalently all isometries). Its elements include translations, rotations about various points, and reflections about various lines.

Children:

- Group action induces homomorphism to the symmetric group
We can view group actions as “bundles of homomorphisms” which behave in a certain way.

- Orbit-stabiliser theorem
The Orbit-Stabiliser theorem tells us a lot about how a group acts on a given element.

- Stabiliser is a subgroup
Given a group acting on a set, each element of the set induces a subgroup of the group.

- Group orbits partition
When a group acts on a set, the set falls naturally into distinct pieces, where the group action only permutes elements within any given piece, not between them.

- Stabiliser (of a group action)
If a group acts on a set, it is useful to consider which elements of the group don’t move a certain element of the set.

Parents:

- Group theory
What kinds of symmetry can an object have?