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Math Help - finite abelian groups, tensor product

  1. #1
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    finite abelian groups, tensor product

    If A and B are finite abelian groups of relatively prime orders, prove that A \otimes_{\mathbb{Z}} B=0. If p is prime and r>s, prove (\mathbb{Z}/p^r\mathbb{Z})\otimes_{\mathbb{Z}}(\mathbb{Z}/p^s\mathbb{Z})=\mathbb{Z}/p^s\mathbb{Z}.

    Any hints would be great (I haven't gotten far with this one!). Thanks in advance.
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  2. #2
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    Quote Originally Posted by riemannsph12 View Post

    If A and B are finite abelian groups of relatively prime orders, prove that A \otimes_{\mathbb{Z}} B=0.
    let |A|=n, \ |B|=m, and a \in A, \ b\in B. since \gcd(n,m)=1, there exist r,s \in \mathbb{Z} such that rn+sm=1. thus: a \otimes b=[(rn+sm)a] \otimes b = (sma) \otimes b = (sa) \otimes mb = (sa) \otimes 0 = 0.


    If p is prime and r>s, prove (\mathbb{Z}/p^r\mathbb{Z})\otimes_{\mathbb{Z}}(\mathbb{Z}/p^s\mathbb{Z})=\mathbb{Z}/p^s\mathbb{Z}.
    see my post in http://www.mathhelpforum.com/math-he...tensor-zn.html for a general form of your question.
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