Mathematics.

group theory

Group Actions

Abstract Algebra I35 minDifficulty6 out of 10

You should know: group mathematics

Overview

A group action is a formal way of describing how a group G 'acts' on a set X by permuting its elements, compatibly with the group's operation. Actions turn abstract group elements into concrete symmetries — rotations acting on points, permutations acting on a set, or a group acting on itself by conjugation — and are the main tool for connecting abstract group structure to combinatorial counting via the orbit-stabilizer theorem. This machinery underlies the proof of the Sylow theorems and Burnside's counting lemma.

Intuition

Think of G as a set of instructions for moving points of X around, where following one instruction and then another is the same as following their combined instruction (that's the compatibility axiom), and doing nothing (e) really does nothing. The orbit of a point x is everywhere x can be sent by the group's moves, while the stabilizer is the subgroup of moves that happen to leave x exactly where it started. The orbit-stabilizer theorem says these two are in perfect balance: a large stabilizer (lots of symmetries fixing x) forces a small orbit, and vice versa — the product of orbit size and stabilizer size is always |G|.

Formal Definition

Definition

A (left) group action of G on a set X is a function G × X → X, written g·x, satisfying:

e⋅x=xfor all x∈Xe \cdot x = x \quad \text{for all } x \in X
Identity acts trivially
g⋅(h⋅x)=(gh)⋅xfor all g,h∈G, x∈Xg \cdot (h \cdot x) = (gh) \cdot x \quad \text{for all } g,h \in G,\ x \in X
Compatibility with the group operation
Orb⁡(x)={g⋅x:g∈G},Stab⁡(x)={g∈G:g⋅x=x}\operatorname{Orb}(x) = \{ g \cdot x : g \in G \}, \qquad \operatorname{Stab}(x) = \{ g \in G : g \cdot x = x \}
Orbit and stabilizer of x
∣Orb⁡(x)∣=[G:Stab⁡(x)]|\operatorname{Orb}(x)| = [G : \operatorname{Stab}(x)]
Orbit-Stabilizer Theorem

Worked Examples

  1. Every symmetry of the square can move any corner to any other corner, so the orbit of a corner is all 4 corners.

    Orb⁡(x)=X, ∣Orb⁡(x)∣=4\operatorname{Orb}(x) = X,\ |\operatorname{Orb}(x)| = 4

  2. By orbit-stabilizer, |Stab(x)| = |G|/|Orb(x)| = 8/4 = 2.

    ∣Stab⁡(x)∣=8/4=2|\operatorname{Stab}(x)| = 8/4 = 2

  3. Indeed, the identity and the reflection through the diagonal at that corner are the only symmetries fixing it.

    Stab⁡(x)={e,diagonal reflection}\operatorname{Stab}(x) = \{e, \text{diagonal reflection}\}

Answer: The orbit is all 4 corners, and the stabilizer has order 2 — {identity, one diagonal reflection} — confirming 4 × 2 = 8 = |D₄|.

Practice Problems

Difficulty 5/10

A group G of order 24 acts on a set X, and one element x ∈ X has stabilizer of order 6. What is |Orb(x)|?

Difficulty 5/10

Verify the two group action axioms for the action of (ℤ, +) on ℝ by translation: n·x = x + n.

Difficulty 6/10

In the conjugation action of a group G on itself, show that Stab(x) is the centralizer C_G(x), and that Orb(e) = {e}.

Quiz

The orbit-stabilizer theorem relates orbit size, stabilizer size, and group order by:
In the conjugation action of G on itself, the stabilizer of an element x is:
A group action of G on X must satisfy e·x = x and:

Summary

  • A group action G × X → X satisfies e·x = x and g·(h·x) = (gh)·x, turning abstract elements into concrete symmetries of X.
  • Orbit-Stabilizer Theorem: |Orb(x)| = [G:Stab(x)], so |Orb(x)|·|Stab(x)| = |G| for finite G.
  • Conjugation of G on itself gives orbits = conjugacy classes and stabilizers = centralizers, the key tool behind the Sylow theorems and class equation.

References

  1. WebsiteWikipedia — Group action