We return to group theory, now armed with the machinery of group actions (introduced in Lecture 3). Understanding what sets a group acts on—and how—reveals the internal structure of itself. The main applications are the class equation, the Sylow theorems, and the classification of finite subgroups of .
11.1.1 Recap and Definitions
Recall from Lecture 3: an action of a group on a set is a homomorphism . Equivalently, a map , , satisfying and for all , .
Definition 11.1.1 (Faithful action). An action is faithful if is injective. Otherwise, the group that “really” acts on is .
Definition 11.1.2 (Orbit). The orbit of under is .
The relation with is an equivalence relation on : reflexivity from , symmetry from , and transitivity from . The orbits are the equivalence classes, giving a partition .
Definition 11.1.3 (Transitive action). An action is transitive if there is only one orbit, i.e., for all there exists with .
Note. Any -action on restricts to a transitive -action on each orbit. So every group action decomposes into a disjoint union of transitive actions.
Definition 11.1.4 (Stabilizer and fixed points).
- The stabilizer of is . This is a subgroup of .
- The fixed points of are the subset .
Proposition 11.1.1 (Conjugate stabilizers). If , then . In particular, elements in the same orbit have conjugate stabilizers.
Proof. If , then , so . The reverse inclusion follows by applying the same argument with . ∎
11.1.2 The Orbit-Stabilizer Theorem
Example 11.1.1 (The coset action). Given a subgroup , the set of left cosets carries a -action by left multiplication: . This action is transitive ( maps to ). The stabilizer of is itself, and .
The coset action is the universal example of a transitive action:
Theorem 11.1.1 (Orbit-Stabilizer Theorem). If acts on , let and . Then the map is a bijection that intertwines the -actions: .
Proof. Well-defined: if , then .
Surjective: by definition of orbit, every element of is for some .
Injective: .
Equivariance: . ∎
Corollary 11.1.1 (Counting formula). If and are finite, then and .
Example 11.1.2 (Rotational symmetries of the tetrahedron). Let be the group of rotational symmetries of a regular tetrahedron, acting on . The action is transitive (any face can be mapped to any other), so . The stabilizer of a face consists of rotations fixing that face: rotations about the axis through the face and the opposite vertex, giving . Thus . In fact : the identity, 8 elements of order 3 (rotations by and about each vertex-to-face axis), and 3 elements of order 2 (rotations by about each edge-midpoint axis, corresponding to etc.).
11.1.3 Burnside’s Lemma
Theorem 11.1.2 (Burnside’s Lemma (Cauchy–Frobenius)). Let be a finite group acting on a finite set . The number of orbits is where is the set of fixed points of . In words: the number of orbits equals the average number of fixed points of elements of .
Proof. Count the set in two ways.
Summing over : .
Summing over : . Group this by orbits. All elements in an orbit have conjugate stabilizers of the same size . So Equating: . ∎
Example 11.1.3 (Coloring a tetrahedron). How many ways to color the faces of a regular tetrahedron with 3 colors, up to rotational symmetry? Here with , and (12 rotations).
- (identity): all 81 colorings are fixed. .
- 8 rotations of order 3 ( about vertex-to-face axes): 3 faces must share a color, the 4th is free. each.
- 3 rotations of order 2 ( about edge-midpoint axes): pairs of faces must match. each.
So .
Note. Burnside’s lemma extends to more complex situations. For coloring the edges of a tetrahedron with 3 colors: .