# Thread: help... direct sum problem

1. ## 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.

2. Originally Posted by deniselim17
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 $\displaystyle pG=0$.