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Thread: isomorphic proofs

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    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,
    $\displaystyle \gcd (n,m)=1$.
    So in your case,
    $\displaystyle \gcd(2,3)=1$
    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, $\displaystyle \mathbb{Z}_6$ 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.
    Prove that Z6 is not isomorphic with S3, although both groups have 6 elements.
    Note that $\displaystyle \mathbb{Z}_6$ is an abelian group. Yet $\displaystyle S_3$ is not (look below and at my other post).
    Attached Thumbnails Attached Thumbnails isomorphic proofs-picture13.gif  
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