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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. –  user3180 Jan 28 '11 at 3:29
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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. –  Rahul Jan 28 '11 at 3:34
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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. –  Arturo Magidin Jan 28 '11 at 4:04
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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! –  Mariano Suárez-Alvarez Jan 28 '11 at 4:29
    
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. –  Arjang Jan 28 '11 at 4:32
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To answer your question "Why should be bother about pure math" (and as a hindsight, just focuss on the application stuff), I always look at the results that were purely abstract w decade ago, but do pop up widely in consumer electronics (like (Goppa-) codes in CDROMs and DVD).

Also, I once answered at my aunts reaction: "But can you buy something at the grocery with it ?" at the presntation of my master's thesis: "No, but wait for a hundred years and a;; has changed...".

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See this answer by Keith Conrad. So it seems that some elementary particles were predicted by group theory before being experimentally discovered. Here is a video of Richard Feynman describing the particle (using "Strangeness minus 3").

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To be precise, the physicists had discovered that there was a correspondence between the bases of irreducible representations of the Lie algebra of certain symmetry groups and certain families of particles. Gell-Mann predicted, based on this theory, the existence of the $\Omega^-$. –  Zhen Lin Jan 28 '11 at 7:34
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In mathematics, we study structures of various sorts, and the relationships (or morphisms) between them, which we usually encode as functions of a special type. More generally, the objects and relationships or morphisms form what is called a category, which consists of the objects and morphisms, and the rule for how morphisms compose with each other.

There are many many important examples of categories, at least one for every branch of mathematics, and often quite a few. (Sets and functions, vector spaces and linear maps, topological spaces and continuous maps, metric spaces and short maps, rings and ring homomorphisms, graphs and graph homomorphisms, and so on.)

If we take any category at all, and examine a single object in it, then the morphisms from that object to itself which are invertible form a group, called the automorphism group of the object. This group abstractly measures the "symmetries" of that object, under the sort of transformations we're considering important. Symmetry has shown itself to be a powerful tool for reasoning in many circumstances -- it allows us to take an argument about one part of a structure and carry it throughout the structure to many other places where it applies.

Because essentially every object we study in mathematics has an automorphism group, it makes sense to study groups in general, so that we don't have to start afresh in each branch of mathematics when we want to consider symmetries of whatever it is that we're studying.

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Group theory is used extensively in chemistry. It's what determines whether or not two molecules will bond (it's based on the symmetry of their orbitals). Group theory is essentially the basis of the Molecular Orbital Theory which is the basis of modern chemistry.

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Group theory (when physicists say this they mean representation theory) is the basis of modern physics. Via Noether's theorem it is the abstract mechanism responsible for conservation laws (e.g. conservation of energy, conservation of momentum) even in classical mechanics. In quantum mechanics, representations are even more important: the representations of a group called $\text{SU}(2)$ describe the difference between bosons and fermions (the difference being their spin, which is the physical property that makes MRI work), and the representations of $\text{SU}(2) \times \text{SU}(2)$ describe the possible orbits of an electron in a hydrogen atom. So group theory can be used, among many many other things, to predict the structure of the periodic table. It is also the foundation of the Standard Model of particle physics.

(Group theory also happens to be fundamental to many modern branches of mathematics, but I figured an application to things that people provably care about outside of mathematics would look better.)

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@PEV: Are you sure that "provably" wasn't what Qiaochu intended? –  Rahul Jan 28 '11 at 18:36
    
"Provably" is what I intended. I think it's safe to argue that people provably care about particle physics. –  Qiaochu Yuan Jan 28 '11 at 19:46
    
Great example, though I wish they would emphasize this aspect more in quantum mechanics classes! –  Elliott Jan 29 '11 at 6:16
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First of all, even phrasing the question as "I don't need group theory if every question that can be solved with group theory can also be solved without it" is misguided. It just so happens that the definition of a group is a natural thing. You might be able to circumvent it sometimes, but that doesn't mean that you should. If a concept naturally suggests itself, then why should one fight hard not to introduce it?

Now, why is it natural? Because there are loads of structures out there in the Platonic world that consist of a set and a binary operation: the integers, the rational numbers, the non-zero rational numbers, matrices, vectors, geometric symmetries (closely related to matrices), permutations,... the list is almost endless. So it makes sense to capture this common feature of so many familiar objects in a definition.

As for applications, traditionally groups were only thought of as symmetries of geometric objects. Even in this narrow context, the abstract framework is useful, e.g. to count solutions to puzzles or ways of colouring a shape. Here is a concrete example of a puzzle that could only be solved and understood in its entirety using abstract groups (since it allowed us to identify a group of symmetries of a certain object with an already familiar symmetry group). It is also mainly in this traditional function, that groups are of paramount importance to physicists.

Of course, the great insight of Galois was that the word "symmetry" shouldn't be understood too narrowly, and since then, groups have completely permeated all of mathematics. Groups describe the complexity of a polynomial, they describe the complexity of a topological space, of an algebraic variety, of a number field, etc. Given a number field, say a Galois extension of $\mathbb{Q}$, pretty much all its important invariants are groups: the Galois group, the class group, the ring of integers, the group of units in the ring of integers, etc. Similarly, given a topological space, you have its fundamental group, higher homotopy groups, homology and cohomology... You can tell your friend that if we didn't have groups, then we wouldn't know how to tell a donut from a ball!

I should probably stop here, since to give a full account of the usefulness of groups, one would have to write a compendium of all of mathematics.

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So if I'm not convinced that groups are important, the way to convince myself would be to learn more math! –  Elliott Jan 29 '11 at 6:24
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@Elliott That would never be a bad idea ;-) –  Alex B. Jan 29 '11 at 7:38
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Check out this link http://plus.maths.org/content/power-groups

I think the subsection titled "Lonely Pursuits with Groups", where they have used the Klein 4-group to decide how many possible end locations are there for the final marble in the game of solitaire is particularly interesting. I was quite stunned at the elegance of the whole thing when I first saw it. I think it is a great example of the power of group theory. It also satisfies the three conditions you have listed.

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That is a nice discussion. For a similar analysis of a board with a different size (fewer holes), see cut-the-knot.org/proofs/PegsAndGroups.shtml. –  KCd Jan 29 '11 at 21:55
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If you care about geometry/topology then you should care at least a little bit about groups. For example, the study of manifolds with a group structure (Lie groups) is a very beautiful and very applicable field. If you can put a group structure on a manifold, you get some topological information for free: e.g. abelian fundamental group, trivial tangent bundle. The applications to physics are huge (c.f. PEV's answer).

Another way group theory comes up in geometry/topology is by assigning a group to a space or an object on the space as a way of measuring something. For example, most of algebraic topology is assigning group invariants to spaces ((co)homology groups, homotopy groups) to "measure holes". This is useful because groups are often easy to distinguish while it is often difficult to directly show that two spaces are not homeomorphic. An example from differential geometry is holonomy: if we are on a manifold with a metric,the holonomy groups quantify how the space is curved.

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"Groups are often easy to distinguish" --- Handle with care! –  Rasmus Jan 28 '11 at 9:43
    
Yes the problem of distinguishing groups can be very difficult (e.g. showing two group presentations are equivalent can be a beeyatch). However, what I meant is that showing directly that there does not exist an isomorphism between two groups is usually easier than showing directly that there is not a homemorphism between two spaces (without appealing to groups or algebraic structures attached to the spaces). –  Eric O. Korman Jan 28 '11 at 19:42
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