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). |

## Other Representations

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). |

## Further Information

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.