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Cayley's Theorem: Any group is isomorphic to a subgroup of a permutations group.
Arthur Cayley was an Irish mathematician. The name Cayley is the Irish name more common spelled Kelly.
Proof:
Let S be the set of elements of a group G and let * be its operation. Let F be the set of onetoone functions from the set S to the set S. Such functions are called permutations of the set. The set F with function composition (·) is a group.
Proof of this proposition:
Function composition is closed and associative. There is an identity element e(x)=x for all x belonging to S. There is an inverse for any function: if f(x)=y then f^{1}(y)=x. Thus (F, ·) is a group.
For any element g of S consider the function f_{g}(x)=g*x for all x in S. This function is an element of F.
Consider f_{g*h}(x). Since G is a group g*h is an element of S and hence f_{g*h} is an element of F. Furthermore, since * is associative in G,
Therefore the set {f_{g} for g in G} is a subgroup of F. G is isomorphic to a subgroup of F with (·).
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