Let $F$ be a field. Determine the possible finite groups $G$ that are isomorphic to a subgroup of $F^+,$ the additive group of $F$.
This problem is a bit confusing for me. $F$ is an arbitrary field, and $F^+$ is an abelian group (finite or infinite). How can I determine the subgroups of a group about which I only know that it's abelian? Does the fact that the set $F-\{0\}$ is a multiplicative group (and other field axioms) tell me anything about $F^+$?
Hint: Let $G$ be a finite subgroup of $F^+$ and let $g \in G$ have order $n$. Consider $ng=0$ as the equation $(n\cdot1_F)\cdot g=0$ in the field $F$.