S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 6: Modules and Group Actions / From my book

B3.2 Group Actions

The notion of a group “acting” on a set underlies much of mathematics. When we study symmetries of geometric objects, permutations of the roots of a polynomial, or even the structure of a group itself, we are studying group actions. Understanding which sets a group acts on, and how, gives information about . These ideas return throughout the book: in Chapter 11 actions are the main tool for analyzing finite groups, and in representation theory groups act on vector spaces by linear maps.

3.2.1 Actions, Orbits, and Stabilizers

Definition 3.2.1 (Group action). A (left) action of a group on a set is a map , written , satisfying:

  1. for all ;
  2. for all and .

We say acts on , or that is a -set.

Recall (Definition 2.2.8) that denotes the group of all bijections under composition.

Proposition 3.2.2 (Actions are homomorphisms to ). Given an action of on , each map , , is a bijection, and is a homomorphism. Conversely, every homomorphism defines an action by .

Proof. By the two axioms, , and likewise . So is a bijection with inverse . The second axiom says , so is a homomorphism. Conversely, if is a homomorphism, then gives , and gives . ∎

So a group action is the same thing as a homomorphism from to a group of permutations. In Chapter 13 we replace by the group of invertible linear maps of a vector space; such homomorphisms are the representations of (Definition 13.1.1).

Definition 3.2.3 (Orbits and stabilizers). Let act on and let .

  1. The orbit of is , also written .
  2. The stabilizer of is .

Proposition 3.2.4 (Stabilizers are subgroups). For any , the stabilizer is a subgroup of .

Proof. We have since . If , then , so . If , then , so . ∎

Proposition 3.2.5 (Orbits partition ). Define if for some . This is an equivalence relation on , and its equivalence classes are the orbits. Hence the orbits form a partition of .

Proof. Reflexivity: . Symmetry: if , then . Transitivity: if and , then . The class of is . ∎

Definition 3.2.6 (Faithful and transitive actions). Let act on , with associated homomorphism .

  1. The kernel of the action is . The action is faithful if is injective, that is, if only acts as the identity on .
  2. The action is transitive if and there is only one orbit: for all there exists with .

If an action is not faithful, the group that “really” acts on is . Any action of on restricts to an action on each orbit, and that action is transitive; so every action is a disjoint union of transitive actions.

Example 3.2.7 (Fundamental examples of group actions).

  1. Permutations: acts on by . This action is faithful (the associated homomorphism is the identity of ) and transitive (the transposition sends to ).
  2. Left multiplication: acts on itself by . This action is faithful, since , and transitive, since . Every stabilizer is trivial.
  3. The coset action: If , then acts on the set of left cosets by . This is well defined: if , then for some , so . The action is transitive, since . The stabilizer of is , because if and only if .
  4. Conjugation: acts on itself by . This action is the subject of Section 3.2.3 below.
  5. Conjugation on subgroups: acts on the set of its subgroups by . (The set is a subgroup, being the image of under the homomorphism of Section 3.2.3.) The subgroups fixed by every are exactly the normal subgroups.
  6. Linear actions: acts on by matrix multiplication, . This is the prototype for representation theory.

Proposition 3.2.8 (Stabilizers along an orbit are conjugate). If , then .

Proof. If , then , so . Since , the same argument gives , that is, . ∎

3.2.2 The Orbit–Stabilizer Theorem

The following theorem is one of the most useful results in group theory. It says that the action of on an orbit looks exactly like the coset action of Example 3.2.7(c), and it connects the size of an orbit to the index of a stabilizer.

Theorem 3.2.9 (Orbit–Stabilizer Theorem). Let act on , let , and let . The map is a well-defined bijection, and it is equivariant: for all . In particular , and if is finite, so the size of every orbit divides .

Proof. Well defined: If , then for some , so .

Injective: If , then , so and .

Surjective: Every element of has the form .

Equivariant: .

The counting statements follow from the bijection and Lagrange’s theorem. ∎

Corollary 3.2.10 (Counting orbits). Let a finite group act on a finite set , and let be one element from each orbit. Then

Proof. The orbits partition (Proposition 3.2.5); now apply the Orbit–Stabilizer Theorem to each orbit. ∎

Example 3.2.11. Consider acting on in the natural way, and take . The orbit is , since the action is transitive. The stabilizer consists of the permutations of , a group isomorphic to , so . Check: .

3.2.3 The Conjugation Action and Conjugacy Classes

A group acts on itself by conjugation, . The axioms hold: , and Write . Each is a homomorphism, since , and a bijection with inverse . So is an automorphism of , and by Proposition 3.2.2 the action is a homomorphism The automorphisms are called inner automorphisms, and their set is a subgroup of .

Definition 3.2.12 (Conjugacy classes and centralizers). Let be a group. Two elements are conjugate if for some . The conjugacy class of is its orbit under conjugation, and the centralizer of is its stabilizer, , the subgroup of elements commuting with .

Proposition 3.2.13 (Properties of conjugacy classes). Let be a group.

  1. The conjugacy classes of form a partition of .
  2. ; if is finite, divides .
  3. if and only if . The kernel of the conjugation action is , so the action is faithful if and only if .
  4. A subgroup is normal if and only if it is a union of conjugacy classes.

Proof. (a) is Proposition 3.2.5, and (b) is the Orbit–Stabilizer Theorem, since .

(c) means , that is, , for all ; this says . An element is in the kernel when for all , that is, when .

(d) If is normal and , then , so is the union of the classes of its elements. Conversely, if is a union of conjugacy classes, then for all and , so is normal by Proposition 3.1.9. ∎

The conjugacy classes of are described in Section 3.4, and Chapter 11 builds on this action to prove the class equation and the Sylow theorems.

3.2.4 Cayley’s Theorem

Theorem 3.2.14 (Cayley’s Theorem). Every group is isomorphic to a subgroup of . If is finite, then is isomorphic to a subgroup of .

Proof. The left multiplication action of on itself (Example 3.2.7(b)) gives a homomorphism , . The action is faithful: if , then . So , is injective, and (Proposition 2.5.6).

If , choose a bijection . Then is an isomorphism , with inverse , since . Composing gives an injective homomorphism . ∎

Note. Cayley’s theorem is simultaneously profound and useless. Profound: every abstract group is “really” a group of permutations, so the group axioms capture exactly the structure of composition of bijections. Useless: the embedding is almost always far larger than necessary. has order , so embedding a group of order into wastes enormous space. Better embeddings come from smaller faithful actions.

The coset action gives such smaller actions. For , the intersection is called the core of in .

Proposition 3.2.15 (Actions on cosets). Let with finite. The action of on gives a homomorphism whose kernel is the core . The core is the largest normal subgroup of contained in : it is normal, it is contained in , and it contains every normal subgroup of contained in . In particular, if contains no normal subgroup of other than , then is isomorphic to a subgroup of .

Proof. The kernel of the action is the set of elements fixing every coset, which is by Example 3.2.7(c). As a kernel it is normal (Theorem 3.1.10), and taking shows it is contained in . If and , then for every , so is contained in the core. For the last statement: the kernel is then , so is injective and is isomorphic to its image (Proposition 2.5.6), a subgroup of . ∎

Example 3.2.16.

  1. Let and . Then , and by the Orbit–Stabilizer Theorem the action on is the natural action of on ; here is an isomorphism.
  2. Let , of order 8, and for a reflection , so . Cayley’s theorem embeds in , but we can do better. From we get , so . Thus , the core of is trivial, and is isomorphic to a subgroup of . (If fixes a vertex of the square, then is the stabilizer of that vertex, and by the Orbit–Stabilizer Theorem this is the action of on the four vertices.)