groups provide a measure of the degree of symmetry in various situations. use of symmetry can simplify many problems. e.g. in euclidean geometry one learns that the medians of any triangle always have a common point. If you realize that the
group of affine transformations of the plane acts on triangles so as to preserve the property of such medians meeting or not meeting, and that every triangle is affine equivalent to an equilateral triangle, then you realize that it is sufficient to prove this intersection property for an equilateral triangle. Then we can use symmetry to prove it in that case. Namely reflection of an equilateral triangle in one median simply interchanges the other two medians. Since the point of intersection of the other two medians must be fixed by their interchange, and all fixed points of the reflection lie on the axis, namely the third median, it follows that all three medians do meet.
the earliest role of group theory i know of is galois's solution of the problem of which polynomial equations in one variable have solution formulas involving only the pure extraction of roots, in addition to the usual arithmetic operations, applied to their coefficients, e.g. as in the quadratic formula. The solution is quite ingenious and involves the gradual enlargment of the field of coefficients by adding in roots of auxiliary equations. Namely if we startt with a field F of coefficients and add in all solutions of some polynomial whose coefficients lie in that field, as well as all elements obtained by the usual arithmetic operations, we obtain an extension "solution" field G. Galois then focused on the arithmetic symmetries of that extension field G, i.e. permutations of its elements, which preserve all arithmetic operations, and also leave fixed all elements of the original coefficient field F. Thus only permutations are allowed which move only the new elements.
Each such field extension can be obtained in stages, where at each stage the group of symmetries obtained is especially simple. Galois showed that if it were true that the large extension field could be obtained by adding in only elements expressible by the pure extraction of roots, and arithmetic operations, then the extension could be obtained by a sequence of simple extensions, in which each symmetry group is abelian. But he also showed that in some cases, there exist equations whose solution field G has a symmetry group that cannot be so obtained, in fact this is true for all general equations of degree 5 or more. Since the symmetry group, or
Galois group, of a general solution field for an equation of degree ≥ 5 cannot be obtained via a sequence of abelian group extensions, it follows that such a solution field cannot be generated by pure root extractions from its coefficient field.
In physics,
E.Noether's theorem apparently states that certain dynamical systems must obey a law of conservation whenever their laws of motion preserve a certain group of symmetries. This apparently let's one use the observed symmetries to predict what quantities will be conserved, but this is not my area of expertise. A precise version appears on page 88 of V. Arnol'd's Mathematical methods of classical mechanics.
Groups of symmetries play a huge role in classifying geometric structures, e.g. in the interplay between
"fundamental groups" of loops in a space, and corresponding groups of translations on their covering spaces. In particular compact topological, as well as Riemann, surfaces fall into three classes according to whether the fundamental group is zero, non zero but abelian, or non abelian. These groups correspond to subgroups of translations, either classical plane translations, or hyperbolic "translations", of their universal covers, namely the sphere, the plane, and the disc. For a three dimensional version of great depth you may google Thurston's geometrization, or hyperbolization, conjectures.
Every geometric mapping of spaces of the same dimension in algebraic geometry, leads to a
"monodromy group", which is a quotient of the fundamental group of the base space acting on fibers of the map. In some important cases these agree with certain Galois groups of field extensions. a non secure link to some of these questions is here:
http://people.math.harvard.edu/~yqzhang/expositions/minor_thesis.pdf
I am told that much of some areas of theoretical physics is concerned with
representations of (Lie) groups.
Since you asked about the zeta function, you may enjoy looking at the book A course in arithmetic, by Serre, where he shows, starting on page 61, how group theory, specifically the
theory of characters of finite groups of form (Z/nZ)*, plays a role in the analytic proof of the theorem on primes in arithemtic progression. Briefly the idea is to show that there is certain uniform symmetry in the way primes ending in different integers are distributed over the whole range of integers. Thus one proves that primes ending in 1, or in 3, or in 7, or in 9, occur symmetrically, hence all these sets of primes must be infinite. I.e. the set of all primes is infinite, and there are essentially the same "number", or same distribution, of primes ending in each of these integers, hence all 4 sets must be infinite. more generally, if m is any positive integer and a is relatively prime to m, then there are an infinite number of primes congruent to a modulo m. Namely one considers
all the numbers a relatively prime to m, these form an abelian group (Z/mZ)*, and one uses symmetry considerations for a corresponding "L-function" similar to the zeta function, to show all possible cases occur with the same distribution.
Another lovely connection between number theory, or arithmetic geometry, and groups, is the fascinating fact that the set of
rational points on a plane cubic curve form a finitely generated abelian group. The question of just which groups can occur is still unknown but I believe it is known which finite cyclic groups can occur in their decomposition.
Since you also mentioned the heat equation, I remark that the Riemann theta function is a kind of fundamental solution of the several variable complex heat equation, and depends on an auxiliary set of variables which determine an an "abelian variety" where the theta function essentially lives, or at least defines a zero divisor of geometric interest. The parameter space of these abelian varieties is a quotient of the Siegel "upper half space", a symmetric space, by the action of a linear group called
the symplectic group. This quotient space serves as a target for the fundamental "Torelli map" which embeds the moduli space of Riemann surfaces into it.
As fresh_42 remarks, the basic results of finite group theory, such as Sylow theorems, do not apply to infinite, in particular linear groups, where the theorems have a different technical flavor. Nonetheless, I understand the question to be about all groups, and as such I am discussing the wide variety of applications of the idea of symmetry, which is shared by finite and infinite groups, indeed which I think unifies their use in all areas.
In this regard, I would perhaps differ in a possibly trivial detail with the first set of examples offered above. The solution of the classical problems of doubling the cube, trisecting the general angle, etc... do not seem to me to be related essentially to symmetry or group theory. Rather their solution depends only on the concept of the degree of a field extension for a solution field of an equation. I.e. the fact that certain solution fields, namely those obtained by geometric constructions, must have degree equal to a power of 2, whereas fields associated to trisecting angles or doubling cubes must have degree divisible by 3, suffices for these problems. There is no question of the computation of the Galois group of the fields, just their degree. I.e. given a finite field extension G over F, one can compute the degree of that extension, and if normal, one can compute the Galois group, a much more sophisticated invariant. Only the first, and simpler invariant, is needed for these classical problems. Thus in my opinion, only field theory, as distinguished from Galois theory is used here, although one frequently finds the opposite statement in books. As to the squaring of the circle, a corollary of the theorem of Lindemann on the transcendence of pi, it may be that symmetry plays some role in the field theoretic arguments for that, but not to my knowledge any group theoretic results per se. I have only glanced at the proof in Niven's book on irrational numbers.this is a long story, endless really. ...
As a personal note, when I interviewed for an honors course in math in college the interviewer asked me to give an example of a group. Since the definition of a group is so trivial, (any set with an associative operation, with an identity and inverses), and yet I too, like the OP, had heard of the enormous depth and importance of group theory, I concluded I must have the wrong definition, and did not say anything for fear of looking like a fool. As a result, I almost missed out on admission to the course.
