Abstract Algebra: Is S(3) Isomorphic to Z(6)?

[oddiseas]

Homework Statement

Is the symmetric group s(3) isomorphic to Z(6), the group of integers modulo six with addition (mod 6) as its binary operation

Homework Equations

Basically i know that the symmetric group is all the different permutations of this set and that there are six of them. I also know that to be isomorphic is must be a one to one and onto map. But i can't figure out how to apply a binary operation between these two groups. In addition is each element of the symmetric group considered an ordering of three distinct points? or is each permutation considered as "1" element.?

The Attempt at a Solution

 

[rasmhop]

Note,
$$(1\, 2)(1\,3) = (1\,3\,2) \qquad (1\,3)(1\,2)=(1\,2\,3)$$
so $S_3$ isn't commutative, but $\mathbb{Z}_6$ is.