# 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?

• The representation theoretic proof that groups of order $p^a q^b$ are solvable uses a bit of Galois theory (at least, the version that I learned did). (It was used in a lemma about averaging roots of unity.) – Lorenzo Apr 10 '16 at 21:35

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.