# isomorphic proofs

• Nov 23rd 2006, 10:45 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.
• Nov 23rd 2006, 10:57 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,
$\displaystyle \mathbb{Z}_n\times \mathbb{Z}_m\simeq \mathbb{Z}_{nm}$ ($\displaystyle n,m\geq 1$)
If and only if,
$\displaystyle \gcd (n,m)=1$.
$\displaystyle \gcd(2,3)=1$
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.
Note that $\displaystyle \mathbb{Z}_6$ is an abelian group. Yet $\displaystyle S_3$ is not (look below and at my other post).