San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

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 fg(x)=g*x for all x in S. This function is an element of F.

Consider fg*h(x). Since G is a group g*h is an element of S and hence fg*h is an element of F. Furthermore, since * is associative in G,

(g*h)*x = g*(h*x) = g*(fh(x)) = fg(fh(x)) = fg·fh(x)
but
(g*h)*x is fg*h(x)
so
fg*h = fg·fh

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


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins