In mathematics, a set is a collection of different things; the things are elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton.

https://en.m.wikipedia.org/wiki/Set_(mathematics)

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.

https://en.m.wikipedia.org/wiki/Group_(mathematics)

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

https://en.m.wikipedia.org/wiki/Group_theory

The study of the development of a concept such as that of a group has certain difficulties. It would be wrong to say that since the non-zero rationals form a group under multiplication then the origin of the group concept must go back to the beginnings of mathematics. Rather we must take the view that group theory is the abstraction of ideas that were common to a number of major areas which were being studied essentially simultaneously.

The three main areas that were to give rise to group theory are:

1. geometry at the beginning of the 19th Century,
2. number theory at the end of the 18th Century,
3. the theory of algebraic equations at the end of the 18th Century leading to the study of permutations.

(1) Geometry has been studied for a very long time so it is reasonable to ask what happened to geometry at the beginning of the 19th Century that was to contribute to the rise of the group concept. Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied. Also the movement to study geometry in n dimensions led to an abstraction in geometry itself. The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet. Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and János Bolyai among others.

Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.

(2) In 1761 Euler studied modular arithmetic. In particular he examined the remainders of powers of a number modulo n. Although Euler's work is, of course, not stated in group theoretic terms he does provide an example of the decomposition of an abelian group into cosets of a subgroup. He also proves a special case of the order of a subgroup being a divisor of the order of the group.

Gauss in 1801 was to take Euler's work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups. He examines orders of elements and proves (although not in this notation) that there is a subgroup for every number dividing the order of a cyclic group. Gauss also examined other abelian groups. He looked at binary quadratic forms

ax²+2bxy+cy² where a,b,c are integers.

Gauss examined the behaviour of forms under transformations and substitutions. He partitions forms into classes and then defines a composition on the classes. Gauss proves that the order of composition of three forms is immaterial so, in modern language, the associative law holds. In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss's works, found a basis for this abelian group.

(3) Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations. Lagrange's main object was to find out why cubic and quartic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x′, x′′, and x′′′. Then, taking 1, w, w² as the cube roots of unity, he examines the expression

R=x′ +wx′′+w²x ′′′

and notes that it takes just two different values under the six permutations of the roots x′,x′′,x′′′. Although the beginnings of permutation group theory can be seen in this work, Lagrange never composes his permutations so in some sense never discusses groups at all.

The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini's work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls permutazione and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely permutazione semplice which are cyclic groups in modern notation, and permutazione composta which are non-cyclic groups. The permutazione composta Ruffini divides into three types which in today's notation are intransitive groups, transitive imprimitive groups and transitive primitive groups.

Ruffini's proof of the insolubility of the quintic has some gaps and, disappointed with the lack of reaction to his paper Ruffini published further proofs. In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.

Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations. However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right. He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term système des substitutions conjuguées for a group. Cauchy calls two permutations similar if they have the same cycle structure and proves that this is the same as the permutations being conjugate.

Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.

Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a proper decomposition if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

Galois' work was not known until Liouville published Galois' papers in 1846. Liouville saw clearly the connection between Cauchy's theory of permutations and Galois' work. However Liouville failed to grasp that the importance of Galois' work lay in the group concept.

Betti began in 1851 publishing work relating permutation theory and the theory of equations. In fact Betti was the first to prove that Galois' group associated with an equation was in fact a group of permutations in the modern sense. Serret published an important work discussing Galois' work, still without seeing the significance of the group concept.

Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations. He defines isomorphism of permutation groups and proves the Jordan-Hölder theorem for permutation groups. Hölder was to prove it in the context of abstract groups in 1889.

Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry. Groups were certainly becoming centre stage in mathematics.

Perhaps the most remarkable development had come even before Betti's work. It was due to the English mathematician Cayley. As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

Cayley's papers of 1854 were so far ahead of their time that they had little impact. However when Cayley returned to the topic in 1878 with four papers on groups, one of them called The theory of groups, the time was right for the abstract group concept to move towards the centre of mathematical investigation. Cayley proved, among many other results, that every finite group can be represented as a group of permutations. Cayley's work prompted Hölder, in 1893, to investigate groups of order

p³,pq²,pqr and p⁴ .

Group theory really came of age with the book by Burnside Theory of groups of finite order published in 1897. The two volume algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra published in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to bring group theory into perhaps the most major theory of 20th Century mathematics.

https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/

Why is group theory important?

Broadly speaking, group theory is the study of symmetry. When we are dealing with an object that appears symmetric, group theory can help with the analysis. We apply the label symmetric to anything which stays invariant under some transformations. This could apply to geometric figures (a circle is highly symmetric, being invariant under any rotation), but also to more abstract objects like functions: x² + y² + z² is invariant under any rearrangement of x, y, and z and the trigonometric functions sin(t) and cos(t) are invariant when we replace t with t+2π.

Conservation laws of physics are related to the symmetry of physical laws under various transformations. For instance, we expect the laws of physics to be unchanging in time. This is an invariance under "translation" in time, and it leads to the conservation of energy. Physical laws also should not depend on where you are in the universe. Such invariance of physical laws under "translation" in space leads to conservation of momentum. Invariance of physical laws under (suitable) rotations leads to conservation of angular momentum. A general theorem that explains how conservation laws of a physical system must arise from its symmetries is due to Emmy Noether.

Modern particle physics would not exist without group theory; in fact, group theory predicted the existence of many elementary particles before they were found experimentally.

The structure and behavior of molecules and crystals depends on their different symmetries. Thus, group theory is an essential tool in some areas of chemistry.

Within mathematics itself, group theory is very closely linked to symmetry in geometry. In the Euclidean plane R², the most symmetric kind of polygon is a regular polygon. We all know that for any n > 2, there is a regular polygon with n sides: the equilateral triangle for n = 3, the square for n = 4, the regular pentagon for n = 5, and so on. What are the possible regular polyhedra (like a regular pyramid and cube) in R³ and, to use a more encompassing term, regular "polytopes" in R^d for d > 3?

• In R³, there are only five (convex) regular polyhedra, called the Platonic solids:
  
• In R⁴, there are only six (convex) regular polytopes.
• For any d > 4, the number of (convex) regular polytopes in R^d is always three: the higher-dimensional analogues of the tetrahedron, cube, and octahedron.

The reason there are only a few regular figures in each R^d for d > 2, but there are infinitely many regular polygons in R², is connected to the possible finite groups of rotations in Euclidean space of different dimensions. Consider another geometric topic: regular tilings of the plane. This means a tiling of the plane by copies of congruent regular polygons, with no overlaps except along the boundaries of the polygons. For instance, a standard sheet of graph paper illustrates a regular tiling of R² by squares (with 4 meeting at each vertex).

Tiling the Plane with Congruent Squares

There are also regular tilings of R² by equilateral triangles (with 6 meeting at each vertex) and by regular hexagons (with 3 meeting at each vertex).

Tiling the Plane with Congruent Equilateral Triangles

Tiling the Plane with Congruent Regular Hexagons

But that is all: there are no tilings of R² by (congruent) regular n-gons except when n = 3, 4, and 6 (e.g., how could regular pentagons meet around a point without overlap?). Furthermore, for these three values of n there is essentially just one regular tiling in each case. The situation is different if we work with regular polygons in the hyperbolic plane H², rather than the Euclidean plane R². The hyperbolic plane H² is the interior of a disc in which the "lines" are diameters passing through the center of the disc or pieces of circular arcs inside the disc that meet the boundary of the disc in 90 degree angles. See the figure below.

Lines in the Hyperbolic Plane H2

In H² there are tilings by (congruent) regular n-gons for any value of n > 2! Here is a tiling of H² by congruent regular pentagons.

Tiling the Hyperbolic Plane with Congruent Regular Pentagons

The figures here are pentagons because their boundaries consists of five hyperbolic line segments (intervals along a circular arc meeting the boundary at 90 degree angles). The boundary parts of each pentagon all have the same hyperbolic length (certainly not the same Euclidean length!), so they are regular pentagons, and the pentagons are congruent to each other in H² even though they don't appear congruent as figures in the Euclidean plane. Thus we have a tiling of H² by congruent regular pentagons with four meeting at each vertex. Nothing like this is possible for tilings of R², the Euclidean plane. There is also more than one tiling of H² by regular n-gons for the same n. For instance, taking n = 3, there is no tiling of H² by congruent equilateral triangles meeting 6 at a vertex, but there are tilings of H² with 7 meeting at a vertex and 8 meeting at a vertex. Here they are.

Tiling the Hyperbolic Plane with Congruent Equilateral Triangles, 7 at a Vertex

Tiling the Hyperbolic Plane with Congruent Equilateral Triangles, 8 at a Vertex

What makes it possible to tile H² by regular polygons in more possible ways than R² is the different structure of the group of rigid motions (distance-preserving transformations) in H² compared to R². The German mathematician Felix Klein gave a famous lecture in Erlangen, in which he asserted that the definition of a geometry is the study of the properties of a space that are invariant under a chosen group of transformations of that space.

Group theory shows up in many other areas of geometry. For instance, in addition to attaching numerical invariants to a space (such as its dimension, which is just a number) there is the possibility of introducing algebraic invariants of a space. That is, one can attach to a space certain algebraic systems. Examples include different kinds of groups, such as the fundamental group of a space. A plane with one point removed has a commutative fundamental group, while a plane with two points removed has a noncommutative fundamental group. In higher dimensions, where we can't directly visualize spaces that are of interest, mathematicians often rely on algebraic invariants like the fundamental group to help us verify that two spaces are not the same.

Classical problems in algebra have been resolved with group theory. In the Renaissance, mathematicians found analogues of the quadratic formula for roots of general polynomials of degree 3 and 4. Like the quadratic formula, the cubic and quartic formulas express the roots of all polynomials of degree 3 and 4 in terms of the coefficients of the polynomials and root extractions (square roots, cube roots, and fourth roots). The search for an analogue of the quadratic formula for the roots of all polynomials of degree 5 or higher was unsuccessful. In the 19th century, the reason for the failure to find such general formulas was explained by a subtle algebraic symmetry in the roots of a polynomial discovered by Evariste Galois. He found a way to attach a finite group to each polynomial f(x), and there is an analogue of the quadratic formula for all the roots of f(x) exactly when the group associated to f(x) satisfies a certain technical condition that is too complicated to explain here. Not all groups satisfy the technical condition, and by this method Galois could give explicit examples of fifth degree polynomials, such as x⁵ - x - 1, whose roots can't be described by anything like the quadratic formula. Learning about this application of group theory to formulas for roots of polynomials would be a suitable subject for a second course in abstract algebra.

The mathematics of public-key cryptography uses a lot of group theory. Different cryptosystems use different groups, such as the group of units in modular arithmetic and the group of rational points on elliptic curves over a finite field. This use of group theory derives not from the "symmetry" perspective, but from the efficiency or difficulty of carrying out certain computations in the groups. Other public-key cryptosystems use other algebraic structures, such as lattices.

Some areas of analysis (the mathematical developments coming from calculus) involve group theory. The subject of Fourier series is concerned with expanding a fairly general 2π-periodic function as an infinite series in the special 2π-periodic functions 1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), and so on. While it can be developed solely as a topic within analysis (and at first it was), the modern viewpoint of Fourier analysis treats it as a fusion of analysis, linear algebra, and group theory.

Identification numbers are all around us, such as the ISBN number for a book, the VIN (Vehicle Identification Number) for your car, or the bar code on a UPS package. What makes them useful is their check digit, which helps catch errors when communicating the identification number over the phone or the internet or with a scanner. The different recipes for constructing a check digit from another string of numbers are based on group theory. Usually the group theory is trivial, just addition or multiplication in modular arithmetic. However, a more clever use of other groups leads to a check-digit construction which catches more of the most common types of communication errors. The key idea is to use a noncommutative group.

On the lighter side, there are applications of group theory to puzzles, such as the 15-puzzle and Rubik's Cube. Group theory provides the conceptual framework for solving such puzzles. To be fair, you can learn an algorithm for solving Rubik's cube without knowing group theory (consider this 7-year old cubist), just as you can learn how to drive a car without knowing automotive mechanics. Of course, if you want to understand how a car works then you need to know what is really going on under the hood. Group theory (symmetric groups, conjugations, commutators, and semi-direct products) is what you find under the hood of Rubik's cube.

https://kconrad.math.uconn.edu/math216/whygroups.html