# Homework Help: Abstract algebra

1. Jan 17, 2010

### oddiseas

1. The problem statement, all variables and given/known data

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

2. Relevant 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 cant 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.?

3. The attempt at a solution

2. Jan 17, 2010

### 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.