1) Do generators exist for S3 (the symmetric group)? I thought the symmetric groups all had generators?

2) When working out products of elements in S3 I get that (231)(213) = (123). This is a product of two 3-cycles and they are not inverses so (123) has to represent a 3-cycle. How do I tell otherwise that this product isn't the identity?

3) When working out the multiplication table I noted several elements that are equivalent, such as (132) and (213). But I can't make this identification lest the table have multiple instances of (132) in a single row of the table. However this implies that the elements of S3 must contain (132)and(213), which makes no sense if |S3| is only 6.

Gaaaaaaaaaaaaaaaaah!!!! This was supposed to be an easy exercise!

-Dan