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.
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 )
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
There is a one-to-one correspondence between group actions of on and homomorphisms .
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 .
- acts on via .
- acts on via matrix multiplication: , where the first juxtaposition stands for the group action definition and the second for matrix multiplication.
A group action is called
- faithful iff ('identity on all elements of enforces identity on ')
- free iff ('different group elements map an to different elements of '), and
- 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 .
A free operation on a nonempty set is faithful.
We now attempt to characterise these three definitions; i.e. we try to find conditions equivalent to each.
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.
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.
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:
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:
Let be a group action and . Then .
First of all, . Let . Then and hence . Further and hence .
Let . If we write for each , then
The following theorem will imply formulas for the cardinalities of , , or respectively.
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.
- 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.
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 .
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.
Let act on itself by conjugation, and let . Then the orbit of is trivial if and only if .
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.
A set together with a group acting on it is an algebraic structure. Hence, we may define some sort of morphisms for those structures.
Let a group act on the sets and . A function is called equivariant iff
We shall now study the following thing:
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 . ∎
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.