## Cayley's Theorem

Cayley's Theorem

If is a group of order then is isomorphic to a subgroup of the permutation group Example: Let be the Klein group of order 4, with every element self inverse, so of order 2.

Example: We can define then is an isomorphism from onto Proof of Cayley's Theorem

Let the elements of be For an element we find the permutation of the elements of formed by multiplying each element on the left by Each must be equal to for some if and only if We must show

1. is a uniquely defined element of for each 2. is one to one.

3. has the morphism property, By construction maps the set to itself. is one to one since if and then and and 2 is proved. Hence is a one to one map of the set to itself , so is a permutation of this set and so 1 is satisfied.

To show 3, we show that and have the same effect on each of the elements in Assume and so that By the definition of if and only if and if and only if by associativity in but if and only if for each hence and have the same effect on every element of so and Cayley's Theorem is proved. 