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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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It's certainly not always true, since it implies for G a finite group that |G| is odd. Is there some missing hypothesis you were given? –  hardmath Jun 10 '11 at 1:22
Exactly, let $\mathbb K$ is a field, which is $\mathbb Q_p$ the $p-$adic,or real field, complex or Laurent series fields. I want to have such partition for the group $(\mathbb K,+)$ –  Hai Minh Jun 10 '11 at 2:04
H D Hung: Assuming the Axiom of choice, provided your field does not have characteristic 2, such a partition always exists. For all the fields you listed it holds, except perhaps the Laurent series field: if it is the Laurent series field with coefficients in a field of characteristic 2, then no; otherwise, then yes. –  Arturo Magidin Jun 10 '11 at 3:25
Thank you, it's very useful to me. –  Hai Minh Jun 10 '11 at 3:44
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1 Answer

up vote 5 down vote accepted

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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Do you mean a $2-$torsion is some $x\neq0$ such that $x+x=0$ ?. So what are examples for these type of such groups? Is it true if $G=(\mathbb K,+)$, where $\mathbb K$ is some field? –  Hai Minh Jun 10 '11 at 2:00
@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 $2$. –  Arturo Magidin Jun 10 '11 at 3:12
Thank you, it's very useful to me. –  Hai Minh Jun 10 '11 at 3:43
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