Abstract Algebra/Group Theory/Group actions on sets
In this section, we will encounter most important application of group theory. This is the notion of an action on some object. Over-generalizing slightly, we only care about groups because they act on things.
There are two equivalent ways to define a group action. We will present both, then prove their equivalence.
Definition 1a: Let be a group and a set. Then a group action (or just action) of on is a binary operation such that for any and any ,
Definition 1b: Let be a group and a set. Then a group action (or just action) of on is a homomorphism .
Theorem 2: Definitions 1a and 1b are equivalent.
Proof: We must show that for each binary operations in Definition 1a there is a unique homomorphism as in Definition 1b and vice versa. Given the binary operation, define the permutation given by . Then for all and . Thus is a homomorphism . Now, given a homomorphism , define a binary operation by . Then and . Thus is a binary operation satisfying the axioms in Definition 1a, and we are done. ∎
Example 3: Let be a group. Then acts on itself by left multiplication, that is, for any .
Example 4: Let be a group with a any subgroup. Then acts on the left cosets by left multiplication, that is, for any .
Definition 5: A set which is acted on by a group is called a -set.
Definition 6: Let be a group acting on a set . Then the action of is said to be
- i) faithfull if is the only element in such that for all , and
- ii) free if is the only element in such that for any element in .
Remark 7: A group acting on itself by left multiplication is a free action. Thus is it also faithfull.
Definition 8: Let be a -set. The action is called transitive if for any , there exists a such that .
Remark 9: We see that the actions in Example 3 and Example 4 are transitive.
Definition 10: Let be a -set and . Define the orbit of to be the set .
Definition 11: Let be a -set and . Define the equivalence relation . (Check that this is an equivalence relation!) Then is called the orbit space of with respect to , and we write .
Remark 12: Since the action is obviously transitive on each orbit, to "understand" group actions we only have to understand transitive actions.
Definition 13: Let be a -set. The stabilizer of is the set .
Lemma 14: Let be a -set and let . Then is a subgroup of .
Proof: Since , . Let . Then , so . Finally, , so . Thus is a subgroup of . ∎
Lemma 15: Let be a -set, , and . Then .
Proof: Let . Then , showing . To show , note that and follow the same arugment, obtaining . ∎
A -set is an algebraic structure, and as in any situation where we have several instances of a structure, we cannot resist the temptation to introduce maps between them.
Definition 16: Let and be -sets with binary operations and respectively. Then a function is called an equivariant function if for all . is called an isomorphism if is bijective.
Lemma 17: Compositions of equivariant functions are equivariant.
Proof: Trivial. ∎
Theorem 18: Every transitive action of on a set is isomorphic to left multiplication on , where for any .
Proof: By Lemma 15, the stabilizers of any two elements in an orbit are conjugate and so isomorphic. Thus the chosen element is arbitrary. Let by for all . Then if , we have , so and and so is well-defined. Now let be defined by . If , then , so and showing that is well-defined. Since and are obviously inverses of each other, this shows that is a bijection. To prove equivariance, observe that . ∎
Corollary 19 (orbit stabilizer theorem): Let be a -set and the orbit of . Then .
Proof: Indeed, from Theorem 18 and Lagrange's theorem we obtain . ∎
Definition 20: The orbit of where is a -set is said to be trivial if . Let denote the set of elements of whose orbits are trivial. Equivalently, .
Lemma 21: Let be a finite set and be a group acting on . Next, let be a set containing exactly one element from each nontrivial orbit of the action. Then, .
Proof: Since the orbits partition , the cardinality of is the sum of cardinalities of the orbits. All the trivial orbits are contained in , so . By Corollary 19, , proving the lemma. ∎
Definition 22: A -group is a group whose order is a finite power of a prime integer .
Corollary 23: Let be a -group acting on a set . Then .
Proof: Since is a -group, divides for each with defined as in Lemma 21. Thus . ∎
Linear group actions on vector spaces are especially interesting. These have a special name and comprise a subfield of group theory on their own, called group representation theory. We will only touch slightly upon it here.
Definition 24: Let be a group and be a vector space over a field . Then a representation of on is a map such that
- i) given by , , is linear in over .
- iii) for all , .
V is called the representation space and the dimension of , if it is finite, is called the dimension or degree of the representation.
Remark 25: Equivalently, a representation of on is a homomorphism . A representation can be given by listing and , .
As a representation is a special kind of group action, all the concepts we have introduced for actions apply for representations.
Definition 26: A representation of a group on a vector space is called faithful or effective if is injective.