## Finite symmetric groups

In the first section we mentioned the
symmetric group *S*_{X} on the set *X*.
In order to prepare further examples and detailed descriptions of actions
we need to consider this group in some detail, in particular for finite *X*.
A first remark shows that it is only the order of *X* which really matters:

**Lemma: **
*
For any two finite and nonempty sets **X* and *Y*,
the natural actions of *S*_{X} on *X* and *S*_{Y} on *Y* are isomorphic if and
only if
* | X | = | Y | *.

This is very easy to check and therefore left as an exercise.
We call * | X | * the
*degree*
of *S*_{X}, of any subgroup *P £S*_{X} and of any * pÎS*_{X}.
In order to examine permutations of degree *n* it therefore suffices to
consider a particular set of order *n* and its symmetric group. For technical
reasons we introduce two such sets of order *n*:

*n:= {0, ...,n-1 } and *__n__:= {1, ...,n },

hoping that it will be always clear from the context if this *set* *n* is meant or its *cardinality* *n*.
It is an old tradition
to prefer the set * *__n__ and its symmetric group which we should
denote by * S*_{n} in order to be consistent.
Hence let
us fix the notation for the elements of * S*_{n} , the corresponding
notation for
the elements of * S*_{n} is then obvious.
A permutation * pÎS*_{n} is written down in full detail by putting the
images * pi* in a row under the points *i Î*__n__, say
This will be abbreviated by
Hence, for example, *S*_{ 3} consists of the following elements:
In our programs permutations will be written in the form *[ p1, p2, ..., pn]*.
There is a program to
compute all elements of the symmetric group * S*_{n} .

last changed: January 19, 2005