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.
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.
Note that is an abelian group. Yet is not (look below and at my other post).Prove that Z6 is not isomorphic with S3, although both groups have 6 elements.