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.

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- November 23rd 2006, 11:45 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. - November 23rd 2006, 11:57 AMThePerfectHacker
Are you familar with the rule that,

( )

If and only if,

.

So in your case,

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, is a cyclic group. The group, is also cyclic because the element is a generator, that is, . 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.