# Thread: finite abelian groups, tensor product

1. ## finite abelian groups, tensor product

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

Any hints would be great (I haven't gotten far with this one!). Thanks in advance.

2. Originally Posted by riemannsph12

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

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