Group theory examples. For students and schoolchildren - books, mathematics, group theory. Brief description of the theory

This text appeared for several reasons. Firstly, the vast majority have no idea what modern mathematics does. Group theory is, of course, not all of modern mathematics, but only a small part of it, but it is one of the most high levels abstraction, which makes it a good example of a branch of modern mathematics.

Secondly, such a natural and simple (to explain) object as a group is practically unknown to most scientists. Indeed, what could be more natural and familiar to a person than the concept of symmetry. From birth, we, voluntarily or involuntarily, look for symmetry in surrounding objects, and the more symmetrical the object, the more perfect it seems to us. The ancient Greeks considered the ball perfect figure, precisely because the ball has a lot of symmetries. Take a look at any famous painting and you will see a clear axis (and sometimes more than one) of symmetry. Any piece of music develops in a cycle, constantly returning to the original theme, i.e. there is symmetry there too. Even such a well-known symbol as the cross, revered in many religions, seems beautiful to us because of the large number of symmetries: it can be rotated and reflected in relation to any of its parts. But turn the cross into a swastika, and you will immediately have an uncomfortable feeling, because you have destroyed most of the symmetries of the cross. Thus, it is symmetry that determines how perfect a particular object seems to us, and group theory, as a science that studies symmetries, can, without exaggeration, be called the science of perfection.

And thirdly, I am inspired by the example of such wonderful scientists and popularizers of science as Sergei Popov and Igor Ivanov, whose popular science articles I read with interest.

Since the text was originally intended to be accessible to the reader, knowledgeable in mathematics in volume school curriculum, some special parts of the text (in fact, the vast majority of it), containing material more difficult to understand than is usually given in a school algebra course, will begin with a sign and end with a sign (this does not mean that something is required to understand such text more than school mathematics, difficulties will arise of a logical nature). The fact is that group theory is at one of the highest levels of abstraction in modern mathematics and therefore groups sometimes consist of elements that are very difficult to imagine for an inexperienced reader.

Group theory

Group (mathematics)

Group theory

Basic Concepts

Subgroup Normal subgroup Factor group (semi-)Direct product

Topological

Lie group

Orthogonal group O(n)

Special unitary group SU(n)

G 2 F 4 E 6 E 7 E 8 Lorentz group

Poincaré group

Story

Group theory has three historical roots: the theory of algebraic equations, number theory and geometry. The mathematicians who stood at the origins of group theory are Leonhard Euler, Carl Friedrich Gauss, Joseph Louis Lagrange, Niels Henrik Abel and Évariste Galois. Galois was the first mathematician to connect group theory with another branch of abstract algebra, field theory, developing the theory now called Galois theory.

One of the first problems that led to the emergence of group theory was the problem of obtaining an equation of degree m, which would have m roots of a given equation of degree n (m< n ). Эту задачу в простых случаях рассмотрел Худде (1659 г.). В 1740 г. Сондерсон заметил, что нахождение квадратичных множителей биквадратных выражений сводится к решению уравнения 6 степени, а Ле Сёр (1748 г.) и Вейринг (с 1762 по 1782 гг.) развили эту идею.

The general basis for the theory of equations, built on the theory of permutations, in 1770-1771. found Lagrange, and on this basis the theory of substitutions subsequently grew. He discovered that the roots of all the resolvents he encountered were rational functions of the roots of the corresponding equations.

Group theory

To study the properties of these functions, he developed the "calculus of combinations" (Calcul des Combinaisons). Contemporary work by Vandermonde (1770) also anticipated the development of group theory.

Paolo Ruffini in 1799 proposed a proof of the unsolvability of equations of the fifth and higher powers in radicals. To prove this, he used concepts from group theory, although he called them by different names. Ruffini also published a letter written to him by Abbati, the leitmotif of which was group theory.

Galois discovered that if an algebraic equation has several roots, then there always exists a group of permutations of these roots such that 1) every function that is invariant under permutations of the group is rational and, conversely, 2) every rational function of the roots is invariant under permutations of the group. He published his first works on group theory in 1829, at the age of 18, but they went virtually unnoticed until his collected works were published in 1846.

Arthur Cayley and Augustin Louis Cauchy were among the first mathematicians to appreciate the importance of group theory. These scientists also proved some of the theory's important theorems. The subject they studied was popularized by Serret, who devoted a section of his book on algebra to the theory, by Jordan, whose Traité des Substitutions became a classic, and by Eugene Netto (1882). ), whose work was translated into English language Cole. Many other mathematicians of the 19th century also made great contributions to the development of group theory: Bertrand, Hermite, Frobenius, Kronecker and Mathieu.

The modern definition of the concept of “group” was given only in 1882 by Walter von Duke.

In 1884, Sophus Lie initiated the study of both transformation groups of what we now call Lie groups and their discrete subgroups; his works were followed by those of Killing, Studi, Schur, Maurer and Elie Cartan. The theory of discrete groups was developed by Klein, Lee, Poincaré and Picard in connection with the study of modular forms and other objects.

In the middle of the 20th century (mainly between 1955 and 1983), a huge amount of work was carried out on the classification of all finite simple groups, including tens of thousands of pages of papers.

Many other mathematicians, such as Artin, Emmy Noether, Ludwig Silow and others, also made significant contributions to group theory.

Brief description of the theory

The concept of a group arose as a result of a formal description of the symmetry and equivalence of geometric objects. In Felix Klein's Erlangen program, the study of geometry was associated with the study of the corresponding groups of transformations. For example, if figures are given on a plane, then a group of movements determines their equality.

Definition . A group is a set of elements (finite or infinite) on which a multiplication operation is specified, which satisfies the following four axioms:

Closedness of a group under the operation of multiplication . For any two elements of a group, there is a third one, which is theirFree group graph of order 2 by work:

Associativitymultiplication operations. The order in which the multiplication is performed is immaterial:

Existence of a single element. There is some element E in the group, the product of which with any element A of the group gives the same element A:

Group theory

Existence of an inverse element. For any element A of the group there is an element A −1 such that their product gives the identity element E:

The axioms of the group do not in any way regulate the dependence of the multiplication operation on the order of the factors. Therefore, generally speaking, changing the order of the factors affects the product. Groups for which the product does not depend on the order of the factors are called commutative or Abelian groups. For an Abelian group

Abelian groups are quite rare in physical applications. Most often, groups that have physical meaning are non-Abelian:

It is convenient to describe finite groups of small size using the so-called. "Multiplication tables". In this table, each row and each column corresponds to one element of the group, and the result of the multiplication operation for the corresponding elements is placed in the cell at the intersection of the row and column.

Below is an example of a multiplication table (Cayley table) for a group consisting of four elements: (1, −1, i, −i) in which the operation is ordinary arithmetic multiplication:

The identity element here is 1, the inverses of 1 and −1 are themselves, and the elements i and −i are the inverses of each other.

If a group has an infinite number of elements, then it is called an infinite group.

When the elements of a group continuously depend on some parameters, then the group is called continuous, or a Lie group. A Lie group is also said to be a group whose set of elements forms a smooth manifold. Using Lie groups as symmetry groups, solutions are found differential equations.

Groups are ubiquitous in mathematics and natural sciences ah, often to detect the internal symmetry of objects (groups of automorphisms). Internal symmetry is usually associated with invariant properties; the set of transformations that preserve this property, together with the operation of composition, form a group called the symmetry group.

IN Galois theory, which gave rise to the concept of a group, groups are used to describe the symmetry of equations whose roots are the roots of some polynomial equation. Because of the important role they play in this theory, solvable groups get their name.

IN algebraic topology groups are used to describe invariants of topological spaces. By invariants we mean properties of space that do not change when it is deformed in some way. Examples of such uses of groups are fundamental groups, homology and cohomology groups.

Lie groups are used in the study of differential equations and manifolds; they combine group theory and mathematical analysis. The field of analysis associated with these groups is called harmonic analysis.

Group theory

In combinatorics, the concepts of permutation groups and group actions are used to simplify the calculation of the number of elements in a set; in particular, Burnside's lemma is often used.

Understanding group theory is also very important for physics and other natural sciences. In chemistry, groups are used to classify crystal lattices and molecular symmetries. In physics, groups are used to describe symmetries that obey physical laws. Particularly important in physics are representations of groups, in particular Lie groups, since they often point the way to “possible” physical theories.

A group is called cyclic if it is generated by one element a, that is, all its elements are powers of a (or, if we use additive terminology, representable in the form na, where n is an integer). Mathematical notation: .

They say that the group acts on a set, if a homomorphism is given from the group

to the group of all permutations of the set . For brevity, it is often written as or.

Examples of groups

The simplest group is the group with the usual arithmetic operation of multiplication, which consists of element 1. Element 1 is the identity element of the group and its inverse:

The next simple example is a group with the usual arithmetic operation of multiplication, which consists of elements (1, -1). Element 1 is the identity element of the group, both elements of the group are inverse to themselves:

The group of the relatively ordinary arithmetic operation of multiplication is a set consisting of four elements (1, -1, i, -i). The identity element here is 1, the inverses of 1 and -1 are themselves, and the elements i and -i are the inverses of each other.

A group is two rotations of space by 0° and 180° around one axis, if the product of two

turns count as their sequential execution. This group is usually designated C 2 . It is isomorphic (that is, identical) to the above group with elements 1 and -1. Rotate at an angle of 0° because it

is identical, designated in the table by the letter E.

Group theory

			R 180
			R 180
	R 180	R 180

The group, together with the identical transformation E, is formed by the inversion operation I, which reverses the direction of each vector. A group operation is the sequential execution of two inversions. This group is usually designated S 2 . It is isomorphic to the above group C2.

By analogy with group C2, it is possible to construct group C3, consisting of rotations of the plane at angles of 0°, 120° and 240°. We can say that the group C 3 is a group of rotations that transform an equilateral triangle into itself.

Elements of group C3

		R 120	R 240
		R 120	R 240
R 120	R 120	R 240
R 240	R 240		R 120

If we add to group C 3 reflections of the triangle relative to its three axes of symmetry (R1, R2, R3), then we get a complete group of operations that transforms the triangle into itself. This group is called

D3.

Elements of group D3

Groups of permutations of roots were studied earlier by others Lagrange and . But the merit of the one who formulated the essential properties of concepts and applied them to the solution of new and difficult tasks. This was done by the French mathematician Galois for the concept of a group. Only after his work did it become a subject of study for mathematicians.

Évariste Galois (1811–1832) was born in Bourg-la-Reine. In 1823, Evariste's parents sent him to study at the Royal College in Paris. Here he became interested in mathematics and began to independently study the works of Legendre, Euler, Lagrange, and Gauss.

Galois is completely captured by Lagrange's ideas. It seems to him, like Abel once, that he has found a solution to a fifth-degree equation. He makes an unsuccessful attempt to enter the Ecole Polytechnique, but his knowledge of the works of Legendre and Lagrange was not enough, and Galois returns to college.

Here happiness smiles on him for the first time - he meets a teacher who was able to appreciate his genius. Richard knew how to rise above official programs, he was aware of the progress of science and sought to broaden the horizons of his students. Richard's comments about Evariste are simple: "He only works in higher fields mathematics".

And indeed, already at the age of seventeen, Galois received his first scientific results. In 1829, his note “Proof of a theorem on periodic continued fractions” was published. At the same time, Galois presented another work to the Paris Academy of Sciences. She got lost at Koshy's.

Galois tries to enter the Polytechnic School a second time, and again fails. To this was soon added an event that shocked the young man: hounded by political opponents, his father committed suicide. The misfortunes that befell Evarist inevitably affected him: he became nervous and hot-tempered.

In 1829, Galois entered the Normal School. It trained candidates for the title of teacher. Here Evariste carried out research on the theory of algebraic equations and in 1830 submitted his work to the competition of the Paris Academy of Sciences. His fate was in the hands of the permanent secretary of the Academy, Fourier. Fourier begins to read the manuscript, but soon dies. The second manuscript, like the first, disappears.

A time came in Galois's life filled with important events. He joined the Republicans, joined the Society of Friends of the People, and enlisted in the artillery of the National Guard. For speaking out against the leadership, he was expelled from the Normal School.

On July 14, 1831, to commemorate the next anniversary of the storming of the Bastille, a demonstration of the Republicans took place. The police arrested many demonstrators, among them Galois. Galois's trial took place on October 23, 1831. He was sentenced to 9 months in prison. Galois continued his research in prison.

On the morning of May 30, 1832, during a duel in the town of Gentilly, Galois was mortally wounded by a bullet in the stomach. A day later he died.

Galois's mathematical works, at least those that survive, amount to sixty small pages. Never before have works of such a small volume brought the author such wide fame.

In 1832, Galois, while in prison, drew up a program that was published only seventy years after his death. But even at the beginning of the twentieth century it did not arouse serious interest and was soon forgotten. Only modern mathematicians, who continued the work of many generations of scientists, finally realized Galois’s dream.

“I beg my judges to at least read these few pages,” Galois began his famous memoir. However, Galois's ideas were so deep and comprehensive that they were truly difficult for any scientist to appreciate at that time.

"...So, I believe that the simplifications obtained by improving calculations (of course, we mean fundamental simplifications, not technical ones) are not at all limitless. The moment will come when mathematicians will be able to foresee algebraic transformations so clearly that that the expenditure of time and paper on their careful carrying out will cease to pay off. I do not say that analysis cannot achieve something new beyond such foresight, but I think that without it one fine day all means will be in vain.

Subject calculations to your will, group mathematical operations, learn to classify them by degree of difficulty, and not by external signs, - these are the tasks of the mathematicians of the future as I understand them, this is the path I want to follow.

Let no one confuse my ardor with the desire of some mathematicians to avoid any calculations at all. Instead of algebraic formulas, they use long arguments and add to the cumbersomeness of mathematical transformations the cumbersomeness of verbal description of these transformations, using a language not suitable for performing such tasks. These mathematicians are a hundred years behind.

Nothing like that happens here. Here I am doing analysis analysis. At the same time, the most complex of the currently known transformations (elliptic functions) are considered only as special cases, very useful and even necessary, but still not general, so refusing further broader research would be a fatal mistake. The time will come when the transformations discussed in the higher analysis outlined here will actually be carried out and will be classified according to the degree of difficulty, and not according to the type of functions arising here."

Here it is necessary to pay attention to the words “group mathematical operations”. Galois undoubtedly means group theory by this.

First of all, Galois was not interested in individual mathematical problems, but in general ideas that determine the entire chain of considerations and guide the logical train of thought. His evidence is based on a deep theory that makes it possible to combine all the results achieved by that time and determine the development of science for a long time to come. Several decades after Galois's death, the German mathematician David Hilbert called this theory "the establishment of a definite framework of concepts." But no matter what name is attached to it, it is obvious that it covers a very large area of \u200b\u200bknowledge.

“In mathematics, as in any other science,” wrote Galois, “there are questions that require solutions precisely in this moment. These are the pressing problems that capture the minds of progressive thinkers regardless of their own will and consciousness."

One of the problems that Évariste Galois worked on was solving algebraic equations. What happens if we consider only equations with numerical coefficients? After all, it may happen that although there is no general formula for solving such equations, the roots of each individual equation can be expressed in radicals. What if this is not the case? Then there must be some kind of sign that allows us to determine whether a given equation is solved in radicals or not? What is this sign?

Galois's first discovery was that he reduced the degree of uncertainty in their meanings, that is, he established some of the “properties” of these roots. The second discovery relates to the method Galois used to obtain this result. Instead of studying the equation itself, Galois studied its “group”, or, figuratively speaking, its “family”.

“A group,” writes A. Dalma, “is a collection of objects that have certain common properties. Let, for example, take real numbers as such objects. General property groups real numbers is that when we multiply any two elements of this group, we also get a real number. Instead of real numbers, movements on a plane studied in geometry can appear as “objects”; in this case, the property of the group is that the sum of any two movements gives again a movement. Moving from simple examples For more complex ones, you can select some operations on objects as “objects”. In this case, the main property of a group will be that the composition of any two operations is also an operation. It was this case that Galois studied. Considering an equation that needed to be solved, he associated a certain group of operations with it (unfortunately, we are not able to clarify here how this is done) and proved that the properties of the equation are reflected in the features of this group. Since different equations can have the same group, it is enough to consider their corresponding group instead of these equations. This discovery marked the beginning of the modern stage in the development of mathematics.

Whatever “objects” the group consists of: numbers, movements or operations, they can all be considered as abstract elements that do not have any specific characteristics. In order to define a group, you only need to formulate general rules, which must be fulfilled in order for a given collection of “objects” to be called a group. Currently, mathematicians call such rules group axioms; group theory consists of listing all the logical consequences of these axioms. At the same time, more and more new properties are consistently discovered; By proving them, the mathematician deepens the theory more and more. It is important that neither the objects themselves nor the operations on them are specified in any way. If after this, when studying some particular problem, it is necessary to consider some special mathematical or physical objects that form a group, then, based on the general theory, it is possible to foresee their properties. Group theory thus provides significant cost savings; In addition, it opens up new possibilities for using mathematics in research."

The introduction of the concept of group freed mathematicians from the burdensome task of considering many different theories. It turned out that you only need to highlight the “main features” of this or that theory, and since, in essence, they are all completely similar, it is enough to designate them with the same word, and it immediately becomes clear that it is pointless to study them separately.

Galois strives to introduce new unity into the expanded mathematical apparatus. Group theory is, first of all, bringing order to mathematical language.

Group theory, starting at the end of the 19th century, had a huge influence on the development of mathematical analysis, geometry, mechanics and, finally, physics. It subsequently penetrated into other areas of mathematics - Lie groups appeared in the theory of differential equations, Klein groups in geometry. Galilean groups in mechanics and groups in the theory of relativity also arose.

All books can be downloaded for free and without registration.

Elliot, Dauber. Symmetry in physics. In 2 volumes. 1983 364+414 pp. djvu. in one archive 7.4 MB.
Two-volume monograph (by English physicists) on the principles of symmetry in physics. Volume 1 briefly outlines the theory of groups and the theory of group representations, which underlies the theory of symmetries, and considers the applications of this theory to the analysis of the structure of atoms and crystal lattices, as well as to the description of the symmetry properties of nuclei and elementary particles. Volume 2 discusses the electronic structure of molecules, symmetry properties of space and time, permutation groups and unitary groups, and properties of particles in external fields.
For a wide range of physicists and mathematicians - researchers, graduate students and students.
The book was written by a physicist and for physicists. This is not a bare abstraction for mathematicians, but a lot has been considered physical systems. I recommend.

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NEW O.V. Bogopolsky. Introduction to group theory. 2002 148 pp. djvu. 732 KB.
The purpose of the book is to provide a quick and in-depth introduction to group theory. The first part sets out the basics of the theory, constructs the Mathieu sporadic group, and explains its connection with coding theory and Steiner systems. The second part examines the Bass-Serre theory of groups acting on trees. A special feature of the book is the geometric approach to the theory of finite and infinite groups. There are a large number of examples, exercises and pictures.
For researchers, graduate students and university students.
This introduction is quite complex and requires good knowledge of algebra.

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OK. Aminov. Theory of symmetry. Lecture notes and assignments. 2002 192 pp. djvu.
This manual is compiled on the basis of the course of lectures “Additional Chapters of Mathematics”, which for many years were read by the author for students specializing in theoretical physics, the elective course “Theory of Symmetry” for third-year students and the course “Additional Chapters of Mathematics with Applications” for undergraduates Faculty of Physics. The content of the lectures is mainly presented in the form of short notes; The topics on which laboratory tasks are performed are described in more detail. Problems for each section are solved by students practical exercises and independently. In general, this manual is intended to help students in extracurricular work with recommended literature.

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V.A. Artamonov, Yu. L. Slovokhotov. Groups and their applications in physics, chemistry, crystallography. 2005 year. 512 pp. djvu. 5.4 MB.
The theory of groups is systematically presented and its physicochemical applications are considered. The basic group constructions, the theory of finitely generated Abelian and crystallographic groups, the foundations of the theory of representations of finite groups, linear groups and their Lie algebras are presented. Quasicrystals, renormalization groups, Hopf algebras and topological groups are briefly discussed. Symmetry relationships in mechanics, molecular spectroscopy, solid state physics, as well as in the theory of atoms, nuclei and elementary particles are discussed.
For students of natural sciences of higher education educational institutions. UMO stamp on classical university education. May be useful for graduate students and researchers.

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Alekseev V. B. Abel’s theorem in problems and solutions. year 2001. 190 pp. PDF. 1.4 MB.
In this book, the reader will learn how to solve 3rd and 4th degree algebraic equations with one unknown and why there are no general formulas (in radicals) for solving higher degree equations. At the same time, he will become acquainted with two very important sections of modern mathematics - group theory and the theory of functions of a complex variable. One of the main goals of this book is to enable the reader to try his hand at mathematics. To do this, almost all the material is presented in the form of definitions, examples and a large number of problems, provided with instructions and solutions.
The book is intended for a wide range of readers interested in serious mathematics (starting with high school students), and does not require the reader to have any special prior knowledge. The book can also serve as a manual for the work of a mathematical circle. I doubt the latter. Now there are no such schoolchildren. But the book is useful.