In Section 1 we considered the set of all mappings . We saw there that the composition of mappings is associative, and that the identity mapping is an identity for composition. However, is not a group, since not every mapping has an inverse, as the next example shows.
As in the previous section, we can hope that the subset of all mappings which do have inverses will form a group. So we want to find out which mappings have inverses. To this end we have to recall certain special kinds of mappings.
If is a finite set of size , then without loss of generality
we may take
In this case we denote by and by .
A mapping can be conveniently written as
a array of originals and their images:
Now we describe another useful way of writing down permutations.
The proof of the above theorem, though not difficult, requires some technical attention, and it would probably not give you deeper insight then a particular example:
You should be able to multiply permutations written as products of cycles,
them into the two-row format.
As for finding inverses,
the following is of help:
Edmund F Robertson
11 September 2006