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More specifically, are there any results in pure, abstract group theory that are most easily proved using Galois theory?

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  • $\begingroup$ 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.) $\endgroup$ – Lorenzo Apr 10 '16 at 21:35
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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.

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