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Thread: help... direct sum problem

  1. #1
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    help... direct sum problem

    I have this Lemma.

    $\displaystyle An\;abelian\;group\;G\;with\;pG={\{0\}}$ $\displaystyle \;is\;a\;vector\;space \;over \;Z_{p},\;$ $\displaystyle and \;it \;is \;a \;direct \;sum \;of \;cyclic \;groups \;of \;order \;p \;when \;G \;is \;finite.$

    I have proven that $\displaystyle G\;is\;a\;vector\;space\;over\;Z_{p}$ and let $\displaystyle the\;basis\;be\;\{ x_{1}, x_{2}, ... , x_{t} \}\;when\;G\;is finite.$

    I'm facing problem in proving G is the direct sum of the $\displaystyle \langle x_{i} \rangle$.

    Thank you.
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  2. #2
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    Quote Originally Posted by deniselim17 View Post
    I have this Lemma.

    $\displaystyle An\;abelian\;group\;G\;with\;pG={\{0\}}$ $\displaystyle \;is\;a\;vector\;space \;over \;Z_{p},\;$ $\displaystyle and \;it \;is \;a \;direct \;sum \;of \;cyclic \;groups \;of \;order \;p \;when \;G \;is \;finite.$

    I have proven that $\displaystyle G\;is\;a\;vector\;space\;over\;Z_{p}$ and let $\displaystyle the\;basis\;be\;\{ x_{1}, x_{2}, ... , x_{t} \}\;when\;G\;is finite.$

    I'm facing problem in proving G is the direct sum of the $\displaystyle \langle x_{i} \rangle$.
    I think you're almost done: because $\displaystyle (x_1,\ldots,x_t)$ is a basis, you have automatically $\displaystyle G=\mathbb{Z}_p x_1\oplus\cdots\oplus \mathbb{Z}_p x_t$, like in any vector space (this is almost the definition of a basis). It remains to see that $\displaystyle \mathbb{Z}_p x = \langle x\rangle$ for every $\displaystyle x$, which is a consequence of the definition ($\displaystyle \langle x\rangle=\mathbb{Z}x$) and of $\displaystyle pG=0$.
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