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Let $G$ be a group with additive operation. Is this true that we could find a subset $P$ of $G\setminus\{0\}$ so that $G\setminus\{0\}=P\cup -P$ and $P\cap -P=\emptyset$. Here we denote $-P=\{-x:\;x\in P\}$.How do we prove it?

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    Thank you, it's very useful to me.2011-06-10

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If your group has $2$-torsion there is no hope for this since any $2$ torsion element would be in $P\cap-P$. On the other hand, if there is no $2$-torsion this is possible, at least assuming the axiom of choice. The group $G\setminus\{0\}$ is partitioned into $2$ element subsets $\{g,-g\}$. For every such pair, choose one to go in $P$ and the other to go in $-P$.

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    @H D Hung: Yes, that is exactly what $2$-torsion means. For the additive group of fields, it holds if and only if the field is not of characteristic $2$.2011-06-10