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A very interesting result in group theory, named for Arthur Cayley, who proved it in 1878. It basically states that every finite group is isomorphic to some permutation group. One might think that this theorem may be very useful in proving whatever result we might want to show about groups, however what it really shows is the generality of groups of permutations. If one is looking for a counterexample for some hypothesis about groups that one wants to disprove, it might be more easily found in a permutation group. For instance, to disprove the converse of Lagrange's theorem, one need look no further than the alternating group A5 of groups of all even permutations which is of order 60. However A5 does not have subgroups of all orders divisible by 60, disproving the conjecture.

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