I know Galois theory is used to study fields using properties of groups. Is it ever used to study groups using properties of fields? More specifically, are there any results in pure, abstract group theory that are most easily proved using Galois theory?

Expanding on the comment of AreaMan, I'd say that character theory is a possible, positive answer. Just check the Wikipedia article to see the role played by algebraic integers- they are indeed a key ingredient in the proof of Burnside's $p^{a} q^{b}$ theorem.
