# Abstract algebra

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 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.?

## Answers and Replies

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.