Let be the subset of consisting of the permutations:

Now show that is a group of permutations.

Now the problem is how do I prove this? Because I know that in order to be a group of permutations I've to show the following properties:

a) The operation of composition of functions qualifies as an operation on the set of all the permutations of

b) This operation is associative.

c) There is a permutation such that and for any permutation of .

d) Finally for every permutation of there is another permutation of such that

and

Do I exhaust all the possible values and show that the above four properties hold true for all combinations?

Is there any easy way to prove that the set of all the above permutations is a group?