isomorphic proofs

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,
$\gcd (n,m)=1$ .
So in your case,
$\gcd(2,3)=1$
The rest is trivial.
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Here is another appoarch. First note that both these groups have the same number of elements, i.e. 6 elements.
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.

Quote:

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

Note that $\mathbb{Z}_6$ is an abelian group. Yet $S_3$ is not (look below and at my other post).