Thread: 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.

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,
( )
If and only if,
.
So in your case,

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, 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.

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

Note that is an abelian group. Yet is not (look below and at my other post).