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Cayley's TheoremWe have stated that on of the main objectives of group theory is to write down a complete list of nonisomorphic groups. At first, such a task appears hopeless. For, as we have seen, groups pop up in some very unexpected places and, therefore, if we set out to compile a list of all nonisomorphic groups, we would hardly begin to know where to look. The following theorem of Cayley solves this dilemma. Theorem 1: Every group is isomorphic to a subgroup of a permutation group Proof: Let G be a group,
_{g}:G G
_{g}(x) = xg^{1} (x G).
If
G S_{G}
defined by (1)
g _{g}
Since What Cayley's theorem tells us is that permutation groups and their subgroups are all the groups that can exist. Unfortunately, the problem of classifying the subgroups of a permutation group is extremely complicated, even in the case of a finite permutation group. Therefore, Cayley's theorem does not allow us to easily identify a complete list of groups. The above argument actually proves somewhat more than claimed. For if G is finite, having order n, then G is isomorphic to a subgroup of S_{G}. Therefore we have
Corollary 2: If G has finite order n, then G is isomorphic to a subgroup of S_{n}.

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