Here we want to relate permutations to other definitions of groups.
|So here we have a simple group, for example, in this case a cyclic group shown as a Cayley diagram:|
|A common method when we want to understand a structure is to look at functions between the structures. In particular 'functors' which are functions which preserve structure.|
|Here we apply one element of the group, in this case '2', known as a representable. This must 'play well' with the internal structure of the group.|
|If we repeat this for all permutations we get a set of permutations (homset).|
The group was represented by a Cayley diagram, in the example above, just to illustrate the situation graphically. Here we do the same example in terms of the group 'multiplication' operation.
|So lets take '2' as a representable then we can see that left 'multiplying' everything by 2 permutes the elements:|
|Again if we repeat this for all permutations we get a set of permutations (homset).|
For more information about functors see this page.
Cayley's Theorem is a special case of the Yonada Lemma (see this page). This extends this idea to other structures. So any structure can be represented as a functor into set provided it is representable.