| The set of all possible permutations of a set of items, under the operation of consecutive application, is a group.
Permutation groups are important in the study of symmetry (the permutation group on n elements is known as the symmetric group Sn), but they are also a way of looking at groups in general: every group can be interpreted as a subgroup of some permutation group (Cayley's theorem).
Proof: the operations of any group G form a set, so we can consider the permutation group P on this set. Consider the mapping f that associates with each operation g in G the permutation in P that maps every a in G to g.a. This mapping is injective: different g produce different f(g). It also preserves the group operation: for all a,b in G, f(a).f(b) = f(a.b). Therefore, the range of f is a group, a permutation group that is isomorphic to G, and a subgroup of P.
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