Abstract Algebra/Group Theory/Group actions on sets

Interesting in it's own right, group actions are a useful tool in algebra and will permit us to prove the Sylow theorems, which in turn will give us a toolkit to describe certain groups in greater detail.



Definition 1.8.1:

Let   be an arbitrary set, and let   be a group. A function


is called group action by   on   if and only if (  denoting the identity of  )

  1.   and
  2.  .

When a certain group action is given in a context, we follow the prevalent convention to write simply   for  . In this notation, the requirements for a group action translate into

  1.   and
  2.  .

There is a one-to-one correspondence between group actions of   on   and homomorphisms  .

Definition 1.8.2:

Let   be a group and   a set. Given a homomorphism  , we may define a corresponding group action by


If we are given a group action  , then


is a homomorphism. The thus defined correspondence between homomorphisms   and group actions   is a bijective one.



Indeed, if   is a homomorphism, then



  is bijective for all  , since


Let also  . Then



We note that the constructions treated here are inverse to each other; indeed, if we transform a homomorphism   to an action via


and then turn this into a homomorphism via


we note that   since  .

On the other hand, if we start with a group action  , turn that into a homomorphism


and turn that back into a group action


then we ended up with the same group action as in the beginning due to  . 

Examples 1.8.3:

  1.   acts on   via  .
  2.   acts on   via matrix multiplication:  , where the first juxtaposition stands for the group action definition and the second for matrix multiplication.

Types of actions


Definitions 1.8.4:

A group action   is called

  1. faithful iff   ('identity on all elements of   enforces identity on  ')
  2. free iff   ('different group elements map an   to different elements of  '), and
  3. transitive iff for all   there exists   such that  .

Subtle analogies to real life become apparent if we note that an action is faithful if and only if for two distinct   there exist   such that  , and it is free if and only if the elements   are all different for all  .

Theorem 1.8.5:

A free operation on a nonempty set is faithful.

Proof:  . 

We now attempt to characterise these three definitions; i.e. we try to find conditions equivalent to each.

Theorem 1.8.6:

A group action   is faithful if and only if the induced homomorphism   is injective.


Let first a faithful action   be given. Assume  . Then for all     and hence  . Let now   be injective. Then .

An important consequence is the following

Corollary 1.8.7 (Cayley):

Every group is isomorphic to some subgroup of a symmetric group.


A group acts on itself faithfully via left multiplication. Hence, by the previous theorem, there is a monomorphism  . 

For the characterisation of the other two definitions, we need more terminology.

Orbit and stabilizer


Definitions 1.8.8:

Let   be a group action, and let  . Then

  •   is called the orbit of   and
  •   is called the stabilizer of  . More generally, for a subset   we define   as the stabilizer of  .

Using this terminology, we obtain a new characterisation of free operations.

Theorem 1.8.9:

An operation   is free if and only if   is trivial for each  .

Proof: Let the operation be free and let  . Then


Since the operation is free,  .

Assume that for each  ,   is trivial, and let   such that  . The latter is equivalent to  . Hence  . 

We also have a new characterisation of transitive operations using the orbit:

Theorem 1.8.10:

An operation   is transitive if and only if for all    .


Assume for all    , and let  . Since   transitivity follows.

Assume transitivity, and let  . Then for all   there exists   with   and hence  . 

Regarding the stabilizers we have the following two theorems:

Theorem 1.8.11:

Let   be a group action and  . Then  .


First of all,  . Let  . Then   and hence  . Further   and hence  . 

Theorem 1.8.12:

Let  . If we write   for each  , then




Cardinality formulas


The following theorem will imply formulas for the cardinalities of  ,  ,   or   respectively.

Theorem 1.8.13:

Let an action   be given. The relation   is an equivalence relation, whose equivalence classes are given by the orbits of the action. Furthermore, for each   the function


is a well-defined, bijective function.



  • Reflexiveness:  
  • Symmetry:  
  • Transitivity:  .


Let   be the equivalence class of  . Then



Let  . Since  ,  . Hence,  . Hence well-definedness. Surjectivity follows from the definition. Let  . Then   and thus  . Hence injectivity. 

Corollary 1.8.14 (the orbit-stabilizer theorem):

Let an action   be given, and let  . Then

 , or equivalently  .

Proof: By the previous theorem, the function   is a bijection. Hence,  . Further, by Lagrange's theorem  . 

Corollary 1.8.15 (the orbit equation):

Let an action   be given, and let   be a complete and unambiguous list of the orbits. Then


Proof: The first equation follows immediately from the equivalence classes of the relation from theorem 1.8.13 partitioning  , and the second follows from Corollary 1.8.14. 

Corollary 1.8.16:

Let an action   be given, let  , and let   be a complete and unabiguous list of all nontrivial orbits (where the orbit of   is said to be trivial iff  ). Then


Proof: This follows from the previous Corollary and the fact that   equals the sum of the cardinalities the trivial orbits. 

The following lemma, which is commonly known as Burnside's lemma, is actually due to Cauchy:

Corollary 1.8.17 (Cauchy's lemma):

Let an action   be given, where   are finite. For each  , we denote .

The class equation


Definition 1.8.18:

Let a group   act on itself by conjugation, i. e.   for all  . For each  , the centraliser of   is defined to be the set


Using the machinery we developed above, we may now set up a formula for the cardinality of  . In order to do so, we need a preliminary lemma though.

Lemma 1.8.19:

Let   act on itself by conjugation, and let  . Then the orbit of   is trivial if and only if  .

Proof:  . 

Corollary 1.8.20 (the class equation):

Let   be a group acting on itself by conjugation, and let   be a complete and unambiguous list of the non-trivial orbits of that action. Then


Proof: This follows from lemma 1.8.19 and Corollary 1.8.16. 

Special topics


Equivariant functions


A set together with a group acting on it is an algebraic structure. Hence, we may define some sort of morphisms for those structures.

Definition 1.8.21:

Let a group   act on the sets   and  . A function   is called equivariant iff


Lemma 1.8.22:



We shall now study the following thing:

Definition 1.8.24:

Let   be a prime number. If   is a group such that   for some  , then   is called a  -group.

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


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.