  Students having had a semester course in abstract algebra are exposed to the
elegant way in which finite group theory leads to proofs of familiar facts in
elementary number theory. In this note we offer two examples of such group
theoretical proofs using the action of a group on a set. The first is Fermat's
little theorem and the second concerns a well known identity involving the
famous Euler phi function. The tools that we use to establish both results are
sometimes seen in a second semester algebra course in which group actions are
studied. Specifically, we will use the class equation of a group action and
Burnside's theorem.
