# The Value and Applications of Group Theory in Mathematics

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In summary, group theory is a young branch of mathematics that was initially driven by curiosity about the structures it reveals. It has since become a useful tool in solving various problems in different fields such as combinatorics, cryptography, linear algebra, crystallography, and number theory. Some specific applications of groups include solving Diophantine equations, classifying finite Abelian groups, and studying automorphism groups in algebraic geometry. In addition, group theory has also played a role in modern physics, such as the fundamental group of manifolds in string theory. While it does not provide a general formula for solving polynomial equations of any degree, it can tell us whether a solution is possible or not. As an undergraduate math student, you may

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Hello there.Questions I have: what is the value of group theory?I am not trying to say that it is not important I want to know what made mathematicians study these objects and we still study them today.I know there are very interesting for me at least examples of groups like the Lie group but some mathematicians study I think generally groups or some types of them.What are some important or interesting applications of them?I know that R equipped with the operation of addition is an abelian group also other sets of numbers are also groups.Other groups I have read I think have to do with similarity of triangles or with squares and perhaps geometrical applications.Could you provide other applications of groups in mathematics perhaps in fields like riemannian geometry, the differential wave equation, other differential equations like the heat equation, the Schrodinger equation, number theoretical topics like the Riemann zeta function?Thank you.

Group theory solved the three classical problems: cube doubling, angle trisection and squaring the circle; plus the great Fermat. It is a comparably young branch of mathematics. I guess many questions came from pure curiosity. You see a structure and it is normal to ask what can be said about it. Finite Abelian groups help to solve Diophantine equations, e.g. the chinese remainder theorem. Other natural questions are about classifications: finite Abelian groups, cyclic groups, permutation groups, and simple finite groups. All of them occur in many places in mathematics: combinatorics, cryptography, linear algebra, crystallography, number theory, We are used to the chain
$$\mathbb{N}\subsetneq\mathbb{Z}\subsetneq\mathbb{Q}\subsetneq\mathbb{R}\subsetneq\mathbb{C}\subsetneq\mathbb{H}\subsetneq\mathbb{O}$$
where the important parts are fields. People could have asked whether there are other fields. What if ##1+\ldots+1=0##? However, I guess it started of with automorphism groups rather than the academic question about fields of characteristic ##\neq 0##.

And if you want to have a modern physical related answer: Poincaré defined a group structure ##1892## as he investigated the fundamental group of manifolds. He did not name it group, yet, but it is the first occurrence.

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universe function said:
I know there are very interesting for me at least examples of groups like the Lie group but some mathematicians study I think generally groups or some types of them.What are some important or interesting applications of them?
The Clebsch-Gordan coefficients are a neat application of Group Theory:

https://en.wikipedia.org/wiki/Clebsch–Gordan_coefficients

Galois: solvability of polynomial equations.

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martinbn said:
Galois: solvability of polynomial equations.
How does Galois theory solve polynomial equations of fifth degree or over?Or does it solve mutivariable polynomials?Like in algebraic geometry?

universe function said:
How does Galois theory solve polynomial equations of fifth degree or over?Or does it solve mutivariable polynomials?Like in algebraic geometry?
It doesn't. It tells you whether it is possible or not.

fresh_42 said:
Group theory solved the three classical problems: cube doubling, angle trisection and squaring the circle; plus the great Fermat. It is a comparably young branch of mathematics. I guess many questions came from pure curiosity. You see a structure and it is normal to ask what can be said about it. Finite Abelian groups help to solve Diophantine equations, e.g. the chinese remainder theorem. Other natural questions are about classifications: finite Abelian groups, cyclic groups, permutation groups, and simple finite groups. All of them occur in many places in mathematics: combinatorics, cryptography, linear algebra, crystallography, number theory, We are used to the chain
$$\mathbb{N}\subsetneq\mathbb{Z}\subsetneq\mathbb{Q}\subsetneq\mathbb{R}\subsetneq\mathbb{C}\subsetneq\mathbb{H}\subsetneq\mathbb{O}$$
where the important parts are fields. People could have asked whether there are other fields. What if ##1+\ldots+1=0##? However, I guess it started of with automorphism groups rather than the academic question about fields of characteristic ##\neq 0##.

And if you want to have a modern physical related answer: Poincaré defined a group structure ##1892## as he investigated the fundamental group of manifolds. He did not name it group, yet, but it is the first occurrence.
How can 1+...+1=0?it can't based on what we define as 1and 0.Could you tell me some examples of applications in number theory and if there are any applications of groups in string theory? I thought I read somewhere something related.I have not read till now about string theory except of some general things.

universe function said:
How can 1+...+1=0?it can't based on what we define as 1and 0.Could you tell me some examples of applications in number theory and if there are any applications of groups in string theory? I thought I read somewhere something related.I have not read till now about string theory except of some general things.
To what level have you studied mathematics? What mathematics do you already know?

martinbn said:
It doesn't. It tells you whether it is possible or not.
You mean for special cases?Does it answer any generally more general formula for example of 7th degree a7x7+...+a1x1+a0=0?Or like a22x2y2+a12x1y2+a21x2y1+a11x1y1+a0 =0 ?

PeroK said:
To what level have you studied mathematics? What mathematics do you already know?
You mean generally?Or in group theory?I am an undergraduate math student but I read on my free time more math in fields like riemannian geometry,manifolds these days I have started geometric analysis but I have a problem on interest because I explore what I read and I want to make questions on the things I read, connect some of them to form new questions I want to make some efforts to try to answer some questions that are on my interest but also accepted by other mathematicians.I have this problem.

universe function said:
You mean generally?Or in group theory?I am an undergraduate math student but I read on my free time more math in fields like riemannian geometry,manifolds these days I have started geometric analysis but I have a problem on interest because I explore what I read and I want to make questions on the things I read, connect some of them to form new questions I want to make some efforts to try to answer some questions that are on my interest but also accepted by other mathematicians.I have this problem.
##1+1=0## in the system you see right in front of your eyes.

universe function said:
You mean for special cases?Does it answer any generally more general formula for example of 7th degree a7x7+...+a1x1+a0=0?
Yes. It is impossible.
Or like a22x2y2+a12x1y2+a21x2y1+a11x1y1+a0 =0 ?
That is algebraic geometry and commutative algebra, not group theory.

fresh_42 said:
##1+1=0## in the system you see right in front of your eyes.
So it is not the 1,0,+ we know from arithmetic I guess.

universe function said:
So it is not the 1,0,+ we know from arithmetic I guess.
It's called binary arithmetic. Computers are quite fond of it.

universe function said:
So it is not the 1,0,+ we know from arithmetic I guess.
No. If we consider all remainders of a division of integers by a fixed given integer, then we a get a ring structure on this set of remainders, i.e. an additive group and a multiplication which is also a group if we divide by a prime. Otherwise we have phenomena like ##2\cdot 3 = 0## if we divide by ##6##.

##\{0,1\}## are the remainders by division by ##2##. It are the bits your cell phone or computer works with. It's also an ordinary light switch, or more generally the TRUE and FALSE in a Boolean algebra: AND is multiplication, XOR is addition.

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fresh_42 said:
No. If we consider all remainders of a division of integers by a fixed given integer, then we a get a ring structure on this set of remainders, i.e. an additive group and a multiplication which is also a group if we divide by a prime. Otherwise we have phenomena like ##2\cdot 3 = 0## if we divide by ##6##.

##\{0,1\}## are the remainders by division by ##2##. It are the bits your cell phone or computer works with. It's also an ordinary light switch, or more generally the TRUE and FALSE in a Boolean algebra: AND is multiplication, OR is addition.
It is a different domain than 1+1=2 otherwise we have a contradiction.

Are there any groups used in differential equations like the examples I said before?

universe function said:
It is a different domain than 1+1=2 otherwise we have a contradiction.
Yes, of course.

universe function said:
Are there any groups used in differential equations like the examples I said before?
Linear groups, yes, finite groups less. But groups operating on tangent bundles is a bit further away from differential equations that the connection could be seen instantly.

You have applications of group theory in SR...
The Galileo group and the Lorentz group and the Poincaré group.
The Galileo group with dimension 6 say us that we have 3 rotation and 3 translations to make in the 3D space.
The Poincaré group, with 4D Minkowsky space, give us 10 dimensions (free "variables", let's say like this). From this 10 dimensions, we have four translations and six rotation.
Now, the six rotations in the 4D Minkowski space of basis (t,x1,x2,x3) gives us:
Three rotation about the t axis, which is, essentially, nothing new.
And the other three rotation continues the 3 rotation of hyperbolic planes (that is, (t,xi)) about others axis (not t, obviously). This rotation forms the Lorentz group.

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.

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Auto-Didact and martinbn
I think there are some distinctions to be made. If someone says group theory he usually refers to finite groups, or maybe infinite ones, but not linear groups, which are the second subject. Partly related to both is representation theory, the third topic. If it comes to physics, then representation theory is normally again about linear groups and / or Lie theory.

Finite groups as far as they are not linear groups are something else. They occur in fields such as listed in post #2. Galileo, Lorentz, Poincaré, Heisenberg and other Lie groups are a different matter. At least I have never seen that the central series of a (generalized) Heisenberg group played any role in physics. The center, yes, but that's it. It's the topology which is important, not the group structure. And even worse: physicists often mean representations of the corresponding Lie algebra if they speak about Lie groups. Hence group theoretical specifica are irrelevant in this area.

fresh_42 said:
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Finite groups as far as they are not linear groups are something else.
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What do you mean by this? Finite groups are linear groups.

martinbn said:
What do you mean by this? Finite groups are linear groups.
A linear group is commonly understood as a subgroup of ##GL(\mathbb{F})##. They are not finite unless ##\mathbb{F}## is finite. E.g. ##\mathbb{Z}_n## is not a linear group, it is a finite, cyclic group. Permutation groups aren't linear either.

fresh_42 said:
A linear group is commonly understood as a subgroup of ##GL(\mathbb{F})##. They are not finite unless ##\mathbb{F}## is finite. E.g. ##\mathbb{Z}_n## is not a linear group, it is a finite, cyclic group. Permutation groups aren't linear either.
Of course, ##GL(\mathbb{F})## has finite subgroups. For example

##\{g\in GL(\mathbb{F}), \text{such that } g=diag(a,a,...,a) \text{ with a root of unity}\}##.

Any finite group is subgroup of a general linear group.

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martinbn said:
Of course, ##GL(\mathbb{F})## has finite subgroups. For example

##\{g\in GL(\mathbb{F}), \text{such that } g=diag(a,a,...,a) \text{ with a root of unity}\}##.

Any finite group is subgroup of a general linear group.
This is misleading and confusing. Every finite group may have a linear representation (proof? a faithful one?), but that doesn't make it a linear group. There is a reason we distinguish groups and their representations. If you introduce your own definition, be it so, but do not pretend as if others do as well. It is a martinbn special to call cyclic groups linear. Nobody else does it. And the symmetric groups aren't linear either.

EDIT: Which linear group is ##\langle a,b,c\,|\,a^5, b^4,c^9,b^3c^7\rangle ##?

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fresh_42 said:
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This is misleading and confusing. Every finite group may have a linear representation (proof? a faithful one?), but that doesn't make it a linear group. There is a reason we distinguish groups and their representations. If you introduce your own definition, be it so, but do not pretend as if others do as well. It is a martinbn special to call cyclic groups linear. Nobody else does it. And the symmetric groups aren't linear either.
This is standard. And in accordance with the definition. A linear group is a subgroup of GL(F). Finite groups are such.

fresh_42 said:
EDIT: Which linear group is ##\langle a,b,c\,|\,a^5, b^4,c^9,b^3c^7\rangle ##?
You want me to write it as a group of matrices? Why?

martinbn said:
This is standard. And in accordance with the definition. A linear group is a subgroup of GL(F). Finite groups are such.
Proof? Again, you confuse representation and group. Please stop spreading misinformation.

martinbn said:
You want me to write it as a group of matrices? Why?
To prove your statement. You didn't quote anything to support your weird use of mathematical language.

I don't understand! Why are you so worked up? The definition, you gave, is: Def: A group is a linear group if it is a subgroup of GL_n(F) for some field F. Then every finite groups is a linear group.

If students were asked in an exam to give an example for a linear group, and they answer ##\mathbb{Z}_4## or ##A_4## then this will get them no points. The theory of finite groups and the theory of linear (algebraic) groups are two different subjects. To mix them isn't helpful. In case we have a homomorphism ##\mathbb{Z}_4 \to GL(V)## it is a linear representation, faithful if the homomorphism is injective, but not a subgroup. Permutation matrices are a linear representation of (alternating) symmetric groups, they do not define the group.

And I want to see the theorem that states that every finite group has a faithful representation in a finite linear group, which is what you presented as common knowledge.

Pardon me for chiming in, but I suspect this is just a matter of differing use of language. Some of us say two groups are the same if they are isomorphic, which is of course a slight exaggeration. So some people consider a group to be linear if it is isomorphic to a subgroup of a general linear group. In the case of the symmetric group on n letters, one can let it act by permuting the column vectors of the identity matrix and get an isomorphic matrix group. I.e. a linear transformation is determined by its action on a basis, and is an isomorphism if it sends a basis to a basis, so we just need to permute the basis vectors. (One can use this as a trick for defining the sign of a permutation in terms of the determinant of its matrix.) Then since any finite group G of order n is isomorphic to a subgroup of the permutation group on n letters (by Cayley's theorem), G is also isomorphic to a subgroup of GL(n). This point of view on linear groups is taken e.g. in wikipedia, but of course not everyone need agree with that usage.

https://en.wikipedia.org/wiki/Linear_group

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fresh_42 said:
And I want to see the theorem that states that every finite group has a faithful representation in a finite linear group, which is what you presented as common knowledge.

Every finite group is (isomorphic to) a subgroup of ##S_n## for some ##n## and ##S_n## has a faithful ##n##-dimensional linear representation (permute the vectors in a given basis).

Edit: Mathwonk said this in his previous post

mathwonk said:
Pardon me for chiming in, but I suspect this is just a matter of differing use of language. Some of us say two groups are the same if they are isomorphic, which is of course a slight exaggeration. So some people consider a group to be linear if it is isomorphic to a subgroup of a general linear group. In the case of the symmetric group on n letters, one can let it act by permuting the column vectors of the identity matrix and get an isomorphic matrix group. I.e. a linear transformation is determined by its action on a basis, and is an isomorphism if it sends a basis to a basis, so we just need to permute the basis vectors. (One can use this as a trick for defining the sign of a permutation in terms of the determinant of its matrix.) Then since any finite group G of order n is isomorphic to a subgroup of the permutation group on n letters (by Cayley's theorem), G is also isomorphic to a subgroup of GL(n). This point of view on linear groups is taken e.g. in wikipedia, but of course not everyone need agree with that usage.

https://en.wikipedia.org/wiki/Linear_group
This may be the case, but I don't think this was the issue. What about PSL? It is usually called a linear algebraic group although by definition it is not in GL, but a quotiont. I have never heard anyone call it isomorphic ot a linear group.