Is there an isomorphism between these two groups?

Question

[Z6,+6] Z6={0,1,2,3,4,5},+6 is addition mod 6.

[S6,o] S6 is a permutation group.

Is there an isomorphism between these two groups?

I have no ideas to figure out isomorphism between such two groups. Thank you!

Answer 1

The more standard notation for the permutation group with six elements would be $S_3$, as there are $6=3!$ permutations of 3 objects. To answer the question suppose,looking for a contradiction, that: $$\phi:\mathbb{Z}_6\rightarrow S_3$$ is a group isomorphism. Since $\mathbb{Z}_6$ is abelian,and $\phi$ is a isomorphism we must have that $S_3$ is abelian. If you could find two permutations of $S_3$ which do not commute then that would contradict the existence of an isomorphism. Depending on what you have covered to write up a full solution you should show that any group isomorphic to an abelian group is abelian. Then explicitly find two elements of $S_3$ which do not commute.