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**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 $\displaystyle \langle x_{i} \rangle$.