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Reordered account of second example to try to explain the motivation for the choice of action in advance rather than with hindsight.
Minor rewording.
I modified the quick description, just to point out that the group that acts is not always the group itself. (So as to accomodate the proof of Cauhcy's theorem.)
I added an example giving a simple proof of Cauchy's theorem, and consequently removed the inline comment about the proof of this theorem.
Added a missing word "divide" in Example 2.
Prettified some links. Removed the request for an example about groups of order pq, since it's now there!
Included another example - I think this is the example being referred to in the 'incomplete' note.
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Tricki