Permutation Groups
A permutation group is a homomorphism from a finite group 
into the symmetric group 
for some 
.
We prefer to think of the symmetric group acting on the right of some element set 
. For example if we take 
then:
To describe such a homomorphism it suffices to give explicitely the action of on some -element set and verify that it satisfies the laws of a group action. That is to say:
Every group acts on itself by right multiplication. If 
then we we have a homomorphism from into . To describe this action explicitely we let 
, Now:
and this clearly satisfies the laws of a group action.
Similarly, every group acts on itself by left multiplication. Again let . Now:
Pay special attention to the role of the inverse when acting on the left.
To see that this truly satisfies the laws of a group action note that:
When we have several different groups acting on the same set, it is convenient to introduce some additional notation, to this end we introduce the following formal definition:
A permutation group is a triple 
where is a finite group, is a finite set and 
is a homomorphism:
Of course if 
then 
. We write 
to indicate the action of a permutation 
on a point 
.
There is a sense in which the left action and the right action of a group on itself are equivalent. To make this sense precise we introduce a second definition:
Two permutation groups 
and 
are said to be permutation equivalent if there is an isomorphism 
and a bijection 
such that for every 
the following diagram commutes:
We can see now by taking:
and 
to be the identity that the left and right actions of a group on itself are equivalent:
Note that
is most definitely not an isomorphism of groups, it is merely a bijection form the set to itself.
As a second example of a permutation group let be any finite group, and let 
be any subgroup. Let 
be a traversal of the cosets of in , with 
. Let:
That is
is the set of right cosets of in . Now acts naturally on the right of :
where
is such that there exists a 
with 
.
Alternatively we could have taken the left coset of in :
Now
acts on the left of 
:
with
exactly as in the previous example.
Note the appearance of the inverse once again when acting on the left. In fact, these two actions are equivalent as can be seen by taking:
and
to be the identity.
We shall see in the next post, that all transitive permutation groups are of this form, and all permutation groups are disjoint orbits of the transitive ones.