I need some help with these,

Prove that Z2 x Z3 is isomorphic with Z6.

Prove that Z6 is not isomorphic with S3, although both groups have 6 elements.

Thank you for any assistance.

Printable View

- Nov 23rd 2006, 10:45 AMPvtBillPilgrimisomorphic proofs
I need some help with these,

Prove that Z2 x Z3 is isomorphic with Z6.

Prove that Z6 is not isomorphic with S3, although both groups have 6 elements.

Thank you for any assistance. - Nov 23rd 2006, 10:57 AMThePerfectHacker
Are you familar with the rule that,

$\displaystyle \mathbb{Z}_n\times \mathbb{Z}_m\simeq \mathbb{Z}_{nm}$ ($\displaystyle n,m\geq 1$)

If and only if,

$\displaystyle \gcd (n,m)=1$.

So in your case,

$\displaystyle \gcd(2,3)=1$

The rest is trivial.

-----

Here is another appoarch. First note that both these groups have the same number of elements, i.e. 6 elements.

Now, $\displaystyle \mathbb{Z}_6$ is a cyclic group. The group, $\displaystyle \mathbb{Z}_2\times \mathbb{Z}_3$ is also cyclic because the element $\displaystyle (1,1)$ is a generator, that is, $\displaystyle <(1,1)>=\mathbb{Z}_2\times \mathbb{Z}_3$. Now by the property of cyclic groups of equal cardinality states they are unique up to isomorphism. Q.E.D.

Quote:

Prove that Z6 is not isomorphic with S3, although both groups have 6 elements.