Someone asked me today, "Why we should care about groups at all?" I realized that I have absolutely no idea how to respond.
One way to treat this might be to reduce "why should we care about groups" to "why should we care about pure math", but I don't think this would be a satisfying approach for many people. So here's what I'm looking for:
Are there any problems that that (1) don't originate from group theory, (2) have very elegant solutions in the framework of group theory, and (3) are completely intractable (or at the very least, extremely cumbersome) without non-trivial knowledge of groups?
A non-example of what I'm looking for is the proof of Euler's theorem (because that can be done without groups).
[Edit] I take back "insolubility of the quintic" as a non-example; I also retract the condition "we're assuming group theory only, and no further knowledge of abstract algebra".