MATH2401 Abstract Algebra (5): Group Actions

This article turns group elements into symmetries of sets. Orbits and stabilizers lead to counting formulas, conjugacy and the class equation, permutation representations, and the classification of finite rotation groups.

Collections of MATH2401 Abstract Algebra notes:

1. Group;
2. Ring;
3. Module;
4. Field;
5. Action (this article);
6. Galois Theory.

Definition. An action of a group on a set is a map , , such that

Then is a -set. The orbit and stabilizer of are

The action is transitive if it has only one orbit.

Proposition. The relation when for some is an equivalence relation; its classes are the orbits. Every stabilizer is a subgroup.

Theorem. Orbit as a coset space. For every action of on and every , the map

is a well-defined -equivariant bijection.

Proof. If , then , so . The reverse implication gives injectivity, the definition of gives surjectivity, and gives equivariance. ◻

Actions on cosets

For , the set of left cosets need not be a group, but it is a -set under . This action is transitive and the stabilizer of is .

Orbit–stabilizer and counting

Theorem. Orbit–stabilizer. For every action and ,

If is finite, this equals .

Corollary. Orbit counting by representatives. If is finite and contains one representative from each orbit, then

When is finite, the final summand is .

Actions on subsets and normalizers

An action on induces an action on its power set by . In particular, acts by conjugation on its set of subgroups.

Definition. The normalizer of is

It is the stabilizer of in the conjugation action.

Proposition. is a subgroup, , and it is the largest subgroup of in which is normal. The number of conjugates of is .

Conjugacy and the class equation

For the conjugation action of on itself, the orbit of is its conjugacy class and its stabilizer is the centralizer

Definition. The center of is

Theorem. Class equation. If is finite and contains one representative of each noncentral conjugacy class, then

Corollary. Every nontrivial finite -group has nontrivial center.

Proof. Every noncentral class size is a power of greater than . The class equation therefore gives . Since , its center has at least elements. ◻

Permutation representations

Theorem. Actions of on correspond bijectively to homomorphisms , by . The kernel is

and is normal in .

Taking with left multiplication recovers Cayley’s theorem.

Finite subgroups of

Theorem. Classification of finite rotation groups. Every finite subgroup of is isomorphic to one of the following types (with the usual low-order coincidences understood):

1. a cyclic group of rotations about one axis;

2. a rotation dihedral group of order , consisting of rotations preserving a regular -gon (the noncyclic elements are half-turns about axes in the polygon’s plane, not plane reflections);

3. the tetrahedral group of order ;

4. the octahedral group of order ;

5. the icosahedral group of order .

Proof outline. The trivial group is , so assume that is nontrivial. For a nonidentity rotation, mark the two poles of its axis and let be the set of all such poles. Double-count pairs with fixing . If represent the pole orbits and , the count yields

Because , this forces or . The two-orbit case gives a common axis and hence . In the three-orbit case, ordering leaves only

The corresponding presentations and orders identify respectively the rotation groups , , , and , realized by the regular polygon, tetrahedron, cube/octahedron, and dodecahedron/icosahedron. ◻