Group Actions, Burnside’s Lemma and the Class Equation — Studies in Algebra and Group Theory

22 Group Actions, Burnside’s Lemma and the Class Equation

24 October 2025 · Denis Auroux

Group actions

(Artin §6.7–6.9)

Definition 22.1. An action of a group on a set is a homomorphism . Equivalently, we have a map such that for all , and .

This generalizes the idea of groups as symmetries of geometric objects. Understanding what sets a group acts on (and in what way) gives info about !

Definition 22.2. An action is faithful if is injective.

(Otherwise, the group that “really” acts on is …)

Definition 22.3. The orbit of under is .

Observe: such that , and then . So the orbits of the -action form a partition of .

Equivalently: the relation such that is an equivalence relation:

since ;
, ; then , so ;
and , and , ; then , hence .

Orbits are the equivalence classes of this relation.

Definition 22.4. An action is transitive if there is only one orbit, i.e. , such that .

Note. Given any -action on , by restriction we get a -action separately on each orbit. Each of these is transitive (by definition), so we can break up any group action into a disjoint union of transitive actions!

Definition 22.5.

The stabilizer of is . This is a subgroup of !
The fixed points of are the subset .

Proposition 22.6. If then . So: elements in the same orbit have conjugate stabilizers.

Proof. , so . Conversely, the same argument for gives , hence equality. ∎

Example 22.7. Given a subgroup , we have a set . To avoid notation confusion, write for elements of .

acts on by left multiplication: . This action is transitive ( maps to ). itself, and .

Claim: this is what a general group action looks like when restricted to an orbit!

Proposition 22.8. If acts on a set , given , let . Then is a bijection, and equivariant, i.e. it intertwines the -actions:

Proof.

Well-defined: if then ✓.
Surjective by definition of the orbit .
Injective: .

∎

I.e. the action of on the orbit is the same as on , and the action of on is obtained as a disjoint union over orbits.

Corollary 22.9. If and are finite,

Example 22.10. Let = the group of rotational symmetries of a tetrahedron, acting on = the set of faces ().

The action is transitive, i.e. there is only one orbit, . The stabilizer of an element = the rotations mapping a face to itself , and so we find .

(In fact : the identity; 8 elements of order 3, the rotations about an axis through a vertex and the center of the opposite face 3-cycles; 3 elements of order 2, the rotations about an axis through the midpoints of two opposite edges etc.)

Burnside’s lemma

Burnside’s lemma = a formula to count orbits of a group action.

Let be a finite group acting on a finite set , and consider Two ways of calculating :

as a sum over : (recall: = fixed points of );
as a sum over : .

But, since all elements in an orbit have conjugate stabilizers, of size as seen above (), we can rewrite this by grouping over orbits: Hence:

Theorem 22.11 (Burnside’s lemma). (the average number of fixed points of elements of ).

Example 22.12. How many ways are there to color the faces of a tetrahedron with 3 colors, up to symmetries?

Show solution

, . , the rotations of the tetrahedron.

= identity: .
rotation (8 such ’s): = colorings where the 3 sides around the axis have the same color .
rotation (3 such ’s): (front/back one color, top/bottom one color).

(Could get this answer by different means… but e.g. coloring the edges of the tetrahedron would get harder without Burnside. Here: .)

Actions of on itself

(Artin §7.1–7.2)

1) acts on itself by left multiplication, .

This is transitive, with , and fixed points . It’s faithful, . So we get:

Theorem 22.13. Every finite group is isomorphic to a subgroup of , .

This is not very useful for understanding , however. A more useful action:

Notes not reached in this lecture, covered at the start of Lecture 24

2) acts on itself by conjugation: acts by .

We’ve seen that this does define a group homomorphism , so it is indeed an action. Now we have a more interesting structure: the orbits of this action are the conjugacy classes in , and the stabilizer of an element is the subgroup of elements which commute with . This is called the centralizer of , .

Note , the center of , is the kernel of the action (i.e. the subgroup of elements which act trivially). So: the action is trivial when is abelian; it is faithful iff .

How does this help?

The conjugacy classes form a partition of , so
For each conjugacy class, divides .
Moreover for the identity element, and iff .

is called the class equation of the group .

This is extremely useful. For example:

Theorem 22.14. If for prime, then must be abelian.

Proof.

Conjugacy classes have order , and . Thus the number of conjugacy classes such that , i.e. of central elements of , must be a multiple of . Hence .
is a subgroup of , so divides : it’s or . If then is abelian!
Now assume , and let . Then commutes with itself and with , so , hence . But is a subgroup of , so . This implies , i.e. commutes with all elements of , i.e. : contradiction. So , is abelian.

∎

(Hence the only groups of order up to isomorphism are and .)

Proposition 22.15. There are exactly 5 groups of order 8 up to isomorphism.

We know the 3 abelian ones: , , . We know = the symmetries of the square. Finally: the quaternion group with (multiplication by flips signs).

Two ways to show there are only two nonabelian groups of order 8:

”By hand” (see the HW hint): suppose and is not abelian.
- Step 1: a group where every element has must be abelian, so there must be an element of order 4 (order 8 would make ).
- Step 2: the order 4 subgroup generated by is normal. Work out the possibilities for multiplication by an element such that .
Using conjugacy and the class equation:
- Step 1: class equation , , has order 2, 4, or 8. abelian. 4 is impossible by the same argument as for above. So .
- Step 2: if then , but . So , and . Hence the class equation is Then work out the possibilities!