S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 3: Group Relations

3.2 Group actions and symmetry

The notion of a group “acting” on a set is one of the main ”substances” which underlies much of our study of mathematics. When we study symmetries of geometric objects, solutions to polynomial equations, or even the structure of groups themselves, we are studying group actions. This section develops the theory systematically—pay close attention, as these ideas will reappear throughout when we discuss representation theory (where groups act on vector spaces via linear maps).

3.2.1 Group actions

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

  1. for all (identity acts trivially);
  2. for all and (compatibility).

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

Note. There is an equivalent, perhaps more fun, formulation: a group action is a homomorphism , where is the group of all bijections . Given an action, define ; the axioms ensure each is a bijection (with inverse ) and that is a homomorphism. Conversely, any such homomorphism defines an action. This perspective is crucial: group actions are the same as representations of into symmetric groups.

Example 3.2.1. Some fundamental examples of group actions:

  1. Left multiplication: acts on itself by . This action is always faithful (only acts trivially) and transitive (any element can be sent to any other).
  2. Conjugation: acts on itself by . The kernel is , the center.
  3. Conjugation on subgroups: acts on the set of its subgroups by . Normal subgroups are precisely the fixed points!
  4. Coset action: If , then acts on (left cosets) by .
  5. Linear actions: acts on by matrix multiplication. This is the prototype for representation theory.

Definition 3.2.2 (Orbits and stabilizers). Let act on . For :

  1. The orbit of is .
  2. The stabilizer (or isotropy group) of is .

Proposition 3.2.1 (Stabilizers are subgroups). For any , the stabilizer is a subgroup of .

Proof that stabilizers are subgroups. Identity: , so . Closure: if , then , so . Inverses: if , then , so . ∎

Proposition 3.2.2 (Orbits partition ). The orbits of a group action form a partition of . Define iff for some ; this is an equivalence relation whose equivalence classes are exactly the orbits.

Proof that orbits partition. Reflexivity: , so . Symmetry: if , then , so implies . Transitivity: if and , then , so . ∎

The following theorem is one of the most useful results in all of group theory. It connects the size of an orbit to the index of a stabilizer.

Theorem 3.2.1 (Orbit-Stabilizer Theorem). Let act on and let . There is a bijection between the orbit and the set of left cosets , given by . In particular, if is finite,

Proof of Orbit-Stabilizer. Define by .

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

Injective: If , then , so , giving .

Surjective: Every element of has the form . ∎

Example 3.2.2. Consider acting on in the natural way. Take . The orbit is (any element can be sent to any other). The stabilizer is , which consists of all permutations of , so and . Check: . ✓

3.2.2 Cayley’s theorem and applications

Theorem 3.2.2 (Cayley’s Theorem). Every group is isomorphic to a subgroup of . If , then embeds into .

Proof of Cayley’s Theorem. Consider the left multiplication action of on itself: . This gives a homomorphism where is the bijection .

Injectivity: If , then for all , . Taking : . Thus , so is injective.

By the First Isomorphism Theorem (cited here ahead of its proof later in this lecture), . If , then . ∎

Note. Cayley’s theorem is simultaneously profound and useless. Profound: every abstract group is “really” a permutation group; the group axioms capture exactly the structure of bijection composition. 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 finding smaller faithful actions. ’Tis fun nonetheless.

Proposition 3.2.3 (Improved Cayley embedding). Let with . The action of on gives a homomorphism with , the largest normal subgroup of contained in .

Proof of improved embedding. The action is well-defined (check!). The kernel consists of all fixing every coset: for all , i.e., for all , i.e., for all . Thus . ∎

Example 3.2.3. Let and (permutations fixing 4). Then , and the action on identifies with the natural action on . The kernel is trivial (since the natural action is faithful), recovering the identity . But: if we take and (stabilizer of a point), then and we get . Since , this is a non-surjective embedding, showing as a proper subgroup of .

Question 2. Let be a group of order where is an odd prime.

  1. Prove that has a unique subgroup of order , which is therefore normal.
  2. Conclude that or (the dihedral group of order ).
Show solution

(1) By Cauchy’s theorem (or direct counting), has an element of order , so is a subgroup of order . Since , we have (index 2 subgroups are always normal: for any , so ).

For uniqueness: suppose is another subgroup of order . Since is prime, is a subgroup of both with order dividing , so . If , then . Otherwise, , and since elements of and (excluding ) have order , we have elements of orders 1 or . But then has only one element not in , call it . We must have (since ), but has order dividing ... this gets complicated. Easier: if , then is a subgroup (since ) of order , contradiction.

(2) Let be the unique subgroup of order . Pick ; then has order 2 (if , then by uniqueness; if , then is cyclic and ).

Assume . Since , conjugation by gives an automorphism of . We have for some . Since : , so , giving .

If : then , so , and is abelian of order with , so .

If : then , i.e., . This is exactly the dihedral relation! We have .