Our mentor just gave us a test, and in it, there was a question that asked the following:

"Show that $\mathbb{Z}_6$ is not isomorphic to $S_3$."

I'm not very sure if I got that one correctly. This is what I did:

"Let $\theta:\mathbb{Z}_6\to S_3$. Notice that $(1\text{ }2)^2=(1\text{ }3)^2=(2\text{ }3)^2=(1)(2)(3)$. Hence, $\theta$ is not one-to-one, and $\mathbb{Z}_6$ is not isomorphic to $S_3$."

Is this okay?

Question

Our mentor just gave us a test, and in it, there was a question that asked the following:

"Show that $\mathbb{Z}_6$ is not isomorphic to $S_3$."

I'm not very sure if I got that one correctly. This is what I did:

"Let $\theta:\mathbb{Z}_6\to S_3$. Notice that $(1\text{ }2)^2=(1\text{ }3)^2=(2\text{ }3)^2=(1)(2)(3)$. Hence, $\theta$ is not one-to-one, and $\mathbb{Z}_6$ is not isomorphic to $S_3$."

Is this okay?

anonymous · Answer

Just for the record,$$S_3=\{(1)(2)(3),(1\text{ }2)(3),(1\text{ }3)(2),(2\text{ }3)(1),(1\text{ }2\text{ }3),(1\text{ }3\text{ }2)\},$$$$\mathbb{Z}_6=\{[0],[1],[2],[3],[4],[5]\}.$$

anonymous · Answer

Does it suffice to say that $S_3 $ is nonabelian while $\mathbb{Z}_6$ is? Else, that looks a correct proof for me.

anonymous · Answer

Hmm, interesting. I will think about that. Thanks! :)

anonymous · Answer

I think you're right. A good example could be:$$(1\text{ }3)(2\text{ }3)=(1\text{ }3\text{ }2)$$$$(2\text{ }3)(1\text{ }3)=(1\text{ }2\text{ }3)$$

anonymous · Answer

That with the fact that $\bigoplus$ is commutative on $\mathbb{Z}_n$ would prove this. Thanks again! :)

anonymous · Answer

No problem. Was afraid my group theory was way too out of touch. Glad to help :-)