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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".

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    Many interesting facts about Rubik's cubes we proven using group theory. In particular, by treating the different permutations of the cube as a finite group and then using Lagrange's Theorem, the maximum number of moves needed to solve the cube was determined. I'll admit this may not be of any practical value, but such information would be hard to find using other methods. The value of Group Theory is in it's generality. The definition of a group is so simple that many real-world problems can give rise to a group, and so much is known about the structure of groups.2011-01-28
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    Why should the insolubility of the quintic be a non-example? Caring about exact solutions to polynomial equations is fairly *concrete* algebra.2011-01-28
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    von Neumann's mathematical work on Quantum Mechanics was based on group theory, and made it much easier to work in QM, as I recall. I'll look for more specific/explicit references tomorrow.2011-01-28
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    Your condition "we're assuming group theory only, and no further knowledge of abstract algebra" is pretty absurd: the relevance of a concept in solving problems is completely uncorrelated to the knowledge of those studying the concept for the first time!2011-01-28
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    Short answer : the consequences of a group definition are useful conceptual structures/frameworks that are used from any where in Crypto Analysis to Quantum Mechanics. But that is their practical use of them. How ever their structure on their own is worth study for those who think group structures are beautiful on their own.2011-01-28
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    Community Wiki? No?2011-08-26
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    Have you ever wondered why there are an infinite number of regular polygons, but only 5 [regular polyhedra](https://en.wikipedia.org/wiki/Platonic_solid)? Symmetry groups help explain this.2015-10-24

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