# isomorphic proofs

• November 23rd 2006, 11:45 AM
PvtBillPilgrim
isomorphic 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.
• November 23rd 2006, 11:57 AM
ThePerfectHacker
Quote:

Originally Posted by PvtBillPilgrim
I need some help with these,
Prove that Z2 x Z3 is isomorphic with Z6.

Are you familar with the rule that,
$\mathbb{Z}_n\times \mathbb{Z}_m\simeq \mathbb{Z}_{nm}$ ( $n,m\geq 1$)
If and only if,
$\gcd (n,m)=1$.
$\gcd(2,3)=1$
Now, $\mathbb{Z}_6$ is a cyclic group. The group, $\mathbb{Z}_2\times \mathbb{Z}_3$ is also cyclic because the element $(1,1)$ is a generator, that is, $<(1,1)>=\mathbb{Z}_2\times \mathbb{Z}_3$. Now by the property of cyclic groups of equal cardinality states they are unique up to isomorphism. Q.E.D.
Note that $\mathbb{Z}_6$ is an abelian group. Yet $S_3$ is not (look below and at my other post).