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The axioms specifying the addition operation of a vector space are precisely those defining an abelian group, so of course it needs to be abelian. That's not sufficient though, as per this quora post..

Obviously the additive group of any field is a vector space over that field or a subfield when viewed as an extension, but is that all? What groups are possible as the additive group of some vector space over some field?

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    $\begingroup$ I think that you should be able to do at least the case when the field has characteristic $p$ yourself. So you're left with asking which abelian groups can be given the structure of a $\mathbb{Q}$-vector space. $\endgroup$ Jun 23 at 8:20

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