Why should we care about groups at all?

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".

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.

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.)

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.