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Cayley's Theorem and its Proof

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 one-to-one 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,

(g*h)*x = g*(h*x) = g*(f_{h}(x)) = f_{g}(f_{h}(x)) = f_{g}·f_{h}(x)
but
(g*h)*x is f_{g*h}(x)
so
f_{g*h} = f_{g}·f_{h}

Therefore the set {f_{g} for g in G} is a subgroup of F. G is isomorphic to a subgroup of F with (·).