## Cayley's Theorem

Cayley's Theorem

Ifis a group of orderthenis isomorphic to a subgroup of the permutation group

Example: Letbe the Klein groupof order 4,with every element self inverse, so of order 2.

Example: We can definethenis an isomorphism fromonto

Proof of Cayley's Theorem

Let theelements ofbeFor an elementwe find the permutation of the elements offormed by multiplying each element on the left byEachmust be equal tofor someif and only if

We must show

1. is a uniquely defined element offor each

2. is one to one.

3. has the morphism property,

By constructionmaps the setto itself.is one to one since ifand   thenandand 2 is proved. Hence is a one to one map of the setto itself , so is a permutation of this set andso 1 is satisfied.

To show 3, we show thatandhave the same effect on each of the elements in

Assumeandso that

By the definition ofif and only ifandif and only ifby associativity inbutif and only iffor eachhenceandhave the same effect on every element ofsoand Cayley's Theorem is proved.