on the set 
.
In order to prepare further examples and detailed descriptions of actions
we need to consider this group in some detail, in particular for finite .
A first remark shows that it is only the order of which really matters:
.
Lemma
For any two finite and nonempty sets and 
,
the natural actions of on and 
on are isomorphic if and
only if
.
This is very easy to check and therefore left as exercise 
.
We call 
the
degree
of , of any subgroup 
and of any 
.
In order to examine permutations of degree 
it therefore suffices to
consider a particular set of order and its symmetric group. For technical
reasons we introduce two such sets of order :
hoping that it will be always clear from the context if this set is meant or its cardinality .
It is an old tradition
to prefer the set 
and its symmetric group which we should
denote by 
in order to be consistent.
Hence let
us fix the notation for the elements of , the corresponding
notation for
the elements of 
is then obvious.
A permutation 
is written down in full detail by putting the
images 
in a row under the points 
, say
This will be abbreviated by
Hence, for example, 
consists of the following elements:
In our programs permutations will be written in the form 
.
There is a program to
compute all elements of the symmetric group .
Exercises
.