# 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.

## Group Actions[edit]

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 ,

- i)

- ii)

**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 sich 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 . ∎

## Group Representations[edit]

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 .

- ii)

- 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.