Then some important concepts in group theory are discussed. Firstly, the operation of group subset is defined and discussed. Based on the subset operation of groups, the concepts and properties of cosets of subgroups are introduced and discussed. The concepts and properties of normal subgroups and quotient groups are defined and discussed. With the help of the concept of quotient group, the basic theorem of group homomorphism is proved and the homomorphism image of group is systematically described. This part is the most basic content in group theory, and it is necessary for any reader who wants to learn group theory. The concept of direct product of groups is given, which is an indispensable tool to study the structure of groups.
Finally, group representation theory's basic theory and application include the concepts of vector space and function space, the direct product of matrix rank sum, invariant subspace and reducible representation, shur lemma, orthogonal theory, characteristic sign, normal function, basis function, direct product of representation and so on.
After the group representation theory, it is its application in quantum mechanics, such as solving some quantum mechanics problems from the perspective of group theory, mainly including the symmetry of Hamiltonian operator, matrix element theorem and selection rule. So as to understand the basic knowledge of group theory and the representation theory of finite groups, and lay the foundation for the application of group theory in physics.
Group theory is one of the great simplification and unification ideas in modern mathematics, and it has important applications in many scientific fields. For example, group theory is the basis of quantum mechanics. It was introduced to understand the solution of polynomial equation, but it was not understood as a mathematical expression of symmetry until the last hundred years. It plays an important role in our understanding of elementary particles, lattice structure and molecular geometry. In this unit, we will study the simple axioms of group satisfaction and begin to develop the basic group theory in an axiomatic way. The purpose of this course is to introduce the concept of group, the concept of axiomatic system, and study the basic properties of groups through examples of group theory, and illustrate these with some important examples, such as general linear groups and symmetric groups.
We give the necessary symbols and basic definitions used in this paper. Firstly, the concepts of subclasses, cosets, problem group decomposition and cosets are defined and discussed. Intersection, and members of double cosets of subclasses. The content of this part is the most basic content, which students must learn.
Representation theory is an important tool to study groups (especially finite groups and compact groups). Broadly speaking, this requires that it is possible to regard a group as a permutation group or a linear group. Many attractive fields of representation theory relate the representation of a group to the representation of its subgroups, especially normal subgroups, algebraic subgroups and local subgroups. Representation theory also considers the images of groups in the automorphism groups of other Abelian groups except simple complex vector space; These are the group modules. This is a more flexible setting than abstract group theory, because we have entered the category of addition. Modular representation theory studies the case that a module is a vector space over a field with positive characteristics.
Finally, the course is about the application of group theory in quantum mechanics. We consider the symmetric operation of the system. Symmetry operation is transformed into Hamiltonian symmetry, which is related to the representation matrix. So there are matrix element theorems and theoretical choices.
Equation theory is the central topic of classical algebra. Until the middle of19th century, algebra was still a mathematical discipline centered on equation theory, and the solution of algebraic equations was still the basic problem of algebra, especially the solution of equations with roots. The so-called equation has a radical solution (algebraic solvability), that is, the solution of this equation is expressed by finite operations such as addition, subtraction, multiplication and division and opening integer idempotent. Group theory also originated from the study of algebraic equations, which is the result of people's logical investigation of solving algebraic equations. Starting with the development of equation theory, this paper expounds the production process of Galois Group Theory and the essence of Galois Theory.
1. the historical background of galois group theory
Judging from the development process of the radical solution of the equation, as early as in the records of ancient Babylonian mathematics and Indian mathematics, they have been able to solve the quadratic equation ax2+bx+c=0 with the radical solution, and the given solution is equivalent to+,which is the square root of the coefficient function. Then the ancient Greeks and ancient orientals solved some special cubic numerical equations, but did not get the general solution of cubic equations. This problem was not solved by the Italians until the heyday of the Renaissance (i.e.1early 6th century). They used Cartan formula to solve the general cubic equation x3+ax2+bx+c=0, where p=ba2 and q=a3. It is obviously obtained by the cubic function of the coefficient. At the same time, Italian Ferrari solved the general quartic equation x4+ax3+bx2+cx+d=0, and used the quartic function of the coefficient to find the root.
16th century has successfully solved the problem of solving equations with roots of quartic and below, but in the following centuries, the general solution of equations with quintic and above has not been obtained. Around 1770, the French mathematician Lagrange reformed the thinking method of algebra, and put forward that the replacement theory of equation roots is the key to solving algebraic equations. By using Lagrange's resolvent method, that is, using any n-degree unit root of 1, the resolvent formula x 1+X2+2 x3+…+N- was derived. His work strongly promoted the development of algebraic equation theory. But his method can't give a radical solution to the general quintic equation, so he suspects that there is no radical solution to the quintic equation. Moreover, he also failed in finding the algebraic solution of the general equation of degree n, thus realizing that the general algebraic equation of degree 4 or more cannot have a radical solution. His thinking method and the method of studying root substitution give inspiration to future generations.
In 1799, Rufini proved that the resolvent of an equation with more than five degrees cannot be less than four, thus proving that the equation with more than five degrees cannot be solved by roots, but his proof is not perfect. In the same year, German mathematician Gauss opened up a new method. In proving the basic theory of algebra, he did not calculate a root, but proved its existence. Later, he began to discuss the concrete solution of higher-order equations. In 180 1, he solved the cyclotomic equation xp- 1=0(p is a prime number), which shows that not all higher-order equations can be solved by roots. Therefore, whether all higher-order equations or some higher-order equations can be solved by roots needs to be further clarified.
Subsequently, the Norwegian mathematician Abel began to solve this problem. From 1824 to 1826, Abel began to study the properties of roots of equations that can be solved by roots, so he corrected the defects in Ruffini's proof and strictly proved that if an equation can be solved by roots, then every root in the expression of roots can be expressed as the root of the equation and the rational number of some unit roots. Abel theorem is proved by this theorem: in general, equations higher than quartic cannot be solved by algebraic method. Then he further thought about which special higher-order equations can be solved by roots. On the basis of the solvability theory of Gauss cyclotomic equation, he solved the solvability problem of an arbitrary special equation, and found that all the roots of this special equation are rational functions of one of the roots (assuming X), and any two roots Q 1(x) and Q2(x) satisfy Q 1Q2 (X) = Q2Q6544. Now this kind of equation is called Abel equation. In fact, some ideas and special results of groups have been involved in the study of Abel equation, but Abel failed to recognize and clearly construct the permutation set of the roots of the equation (because if all the roots of the equation are expressed as rational functions QJ (x 1), J = 1, 2, 3 ..., n, when the other one is used, it should actually be said that the root Xi = Q. Q2(xI), …, Qn(xI) is an arrangement of roots x 1, x2, …, xn), and only the interchangeability Q 1Q2(x)=Q2Q 1(x
Abel solved the problem of constructing algebraic solvable equations of any number, but failed to solve the problem of judging whether known equations can be solved by roots. It is against this background that the French mathematician Galois began to take over Abel's undisputed career.
Two. Galois's work in establishing group theory
Galois carefully studied the predecessors' theories, especially the works of Lagrange, Rufini, Gauss and Abel, and began to study the solvability theory of polynomial equations. He is not in a hurry to find a solution to the higher-order equation, but focuses on judging whether the known equation has a radical solution. If there is, don't ask what the root of the equation is, just prove that there is a root solution.
1. the establishment of galois group theory
Galois, like Lagrange, began with the replacement of the roots of the equation when he proved that there was no general root solution for the equation of degree five or higher. When he systematically studied the permutation and substitution properties of the roots of equations, he put forward some criteria to determine whether the solutions of known equations can be obtained from the roots. However, these methods only made him consider an abstract algebraic theory called "group". In the paper 183 1, Galois first put forward the term "group", called the set of closed permutations a group, and defined the concept of permutation group for the first time. He believes that understanding permutation groups is the key to solving the equation theory, and the equation is a system, and its symmetry can be described by the properties of groups. From then on, he began to solve the problem of equation theory and study group theory directly. He introduced many new concepts about group theory and produced his own Galois group theory, so later people called him the founder of group theory.
Rational coefficient n-degree equation
x+axn- 1+a2xn-2+…+an- 1x+an = 0( 1)，
Suppose every transformation of its n roots x 1, x2, …, xn is called a permutation, and its n roots * * * have n! There are five possible permutations, which are multiplied by the set of permutations to form a group, which is the permutation group of roots. The solvability of the equation can be reflected in some properties of permutation groups of roots, so Galois transformed the solvability of algebraic equations into the analysis of the properties of permutation groups and their subgroups. Now the permutation group associated with the equation (indicating the symmetry of the equation) is called Galois group, which is a group in the coefficient field of the equation. For every polynomial function with a rational function value about the root, the Galois group of the equation is the largest permutation group that meets this requirement. In other words, any polynomial function about the root has a rational value, and every permutation in Galois Group keeps the value of this function unchanged.
2. The essence of Galois group theory
From Galois' work, we can gradually understand the essence of Galois' theory. Firstly, it analyzes how he constructed Galois Group without knowing the root of the equation. It is still the equation (1), and let its roots x 1, x2, ..., xn have no multiple roots. He constructed a Lagrangian resolvent about x 1, x2, ..., xn.
△ 1 = a 1x 1+a2 x2+…+Anxn，
Where ai (I = 1, 2,3, ..., n) is not necessarily a unit root, but it must be some integers, so that n! A linear equation in the form of △ 1 △ 1, △2, …,△n! Different, and then construct an equation.
=0 (2) ,
The coefficients of this equation must be rational numbers (which can be proved by symmetric multinomial theorem) and can be decomposed into the product of irreducible polynomials in the rational number field. Let F(x)= any given irreducible factor of degree m, then the galois group of equation (1) refers to n! All these m's are arranged in a δ i. At the same time, he learned from Vieta's theorem
Knowledge Galois Group is also a symmetric group, which fully embodies the symmetry of the roots of this equation. But it is difficult to calculate the Galois group of a known equation, so the purpose of Galois is not to calculate the Galois group, but to prove that there is always such an equation of degree n, whose Galois group is the possible maximum permutation group S(n) of the root of the equation, and S(n) is composed of n! The product of elements in S(n) actually refers to the product of two permutations. Now the number of elements in S(n) is called order, and the order of S(n) is n! .
After finding the Galois Group G in the coefficient domain of the equation, Galois began to find its largest subgroup H 1. After finding H 1, he used a program containing only rational operations (that is, finding the resolvent) to find a function of the root. The coefficient of belongs to the coefficient domain R of the equation, and its value is not changed under the substitution of H 1, but changed under all other substitutions of G. Then, using the above method, the largest subgroup H2 of H 1 and the largest subgroup H3 of H2 are found in turn, and then H 1, H2, ... At the same time, a series of subgroups and successive resolvents are obtained, and the coefficient domain R is gradually expanded to R 1, R2, …, Rm, and each RI corresponds to a group HI. When Hm=I, Rm is the root domain of the equation, and the rest R 1, R2, …, Rm- 1 are the intermediate domains. Whether an equation can be solved in the form of a root is closely related to the nature of the root domain. For example, quartic equation
x4+px2+q=0 (3)，
P and q are independent, and the coefficient field R is the field obtained by adding letters to rational numbers or unknowns P and Q.. Firstly, calculate its Galois group G, where G is the 8th subgroup of S(4), and G={E, E 1, E2, ... E7}, where
E=，E 1=，E2=，E3=，E4=，E5=，E6=，E7= .
To extend R to R 1, we need to construct a resolvent in R, and then add the root of the resolvent to R to get a new domain R 1, so we can prove that the group of the original equation (3) about the domain R 1 is H 1, H1. Secondly, the second resolvent is constructed to find the root, so the domain R2 is added to the domain R 1, and the group H2 of equation (3) in R2 is found at the same time, where H2={E, E 1}. At this time, the degree of the second resolvent is equal to the exponent of group H2 in H 1. The third step is to construct a third resolvent to find its root, and add it to R2 to get the extension R3. At this time, the group of equation (3) in R3 is H3, H3={E}, that is, H3=I, then R3 is the root domain of equation (3), and the degree of this resolvent is still equal to the exponent of group H3 in H2 21. In this special quartic equation, every time a root is added in the process of expanding the coefficient domain to the root domain, the equation can be solved by a root. This solvable theory is also applicable to general higher-order equations. As long as the root is added every time in the process of extending the coefficient domain to the root domain, the general higher-order equation can also be solved by the root.
Now, taking the quartic equation (3) as an example, Galois found that these resolvents are essentially quadratic binomial equations. Because the solvability principle is also applicable to higher-order equations, for higher-order equations that can be solved by roots in general, its resolvent must exist, and all resolvent formulas should be a binomial equation XP = A. Because Gauss has proved that binomial equations can be solved by roots. So on the contrary, if all the successive resolvents of any higher-order equation are binomial equations, then the original equation can be solved by roots. Therefore, Galois introduced the principle of radical solution and an important concept in group theory, "normal subgroup".
His definition of normal subgroup is as follows: Let H be a subgroup of G, and if every G in G has gH=Hg, then H is a normal subgroup of G, where gH refers to a new permutation set obtained by replacing G first and then applying any element of H, that is, multiplying any element of G by all permutations of H. After the introduction of the definition, Galois proved that when a group is reduced as a reduction equation (such as from G to H 1) On the other hand, if H 1 is a normal subgroup of G and the exponent is a prime number p, then the corresponding resolvent must be a binomial equation of degree p. He also defined a maximal normal subgroup: if a finite group has a normal subgroup, there must be a subgroup whose order is the largest of all normal subgroups in the finite group, and this subgroup is called the maximal normal subgroup of the finite group. A maximal normal subgroup has its own maximal normal subgroup, and this sequence can continue one after another. So any group can generate a maximal normal subgroup sequence. He also proposed that if the sequence of maximal normal subgroups generated by a group G is labeled as G, H, I, J…, then the synthesis factors of a series of maximal normal subgroups can be determined as [G/H], [H/I], [I/G]…. Comprehensive factor [g /H] = order of g/order of h For the quartic equation (3) above, H 1 is a maximal normal subgroup of g, H2 is a maximal normal subgroup of H 1, and H3 is a maximal normal subgroup of H2, that is, a maximal normal subgroup g, h/kloc is generated for the group of equation (3).
With the deepening of the theory, Galois found that for a given equation, finding its sequence in Galois Group and its maximal invariant subgroups is completely a matter of group theory. So he used the method of group theory to solve the solvability problem of the equation completely. Finally, Galois put forward another important concept of group theory, "solvable group". He said that a group is solvable if all the maximal normal composition factors generated by it are prime numbers.
According to Galois theory, if all the largest normal synthesis factors generated by Galois group are prime numbers, the equation can be solved by roots. If it is not a total prime number, it cannot be solved by radical formula. Due to the introduction of solvable groups, it can be said that an equation can be solved by roots if and only if the groups in the coefficient domain of the equation are solvable groups. For the above special quartic equation (3), [G/H]=8/4=2, [H 1/H2]=2/ 1=2, and 2 is a prime number, so equation (3) can be solved by roots. Looking at the general equation of degree n again, when n=3, there are two quadratic resolvent formulas t2=A and t3=B, and the synthetic sequence index is 2 and 3, both of which are prime numbers, so the general cubic equation can be solved by roots. Similarly, for n=4, there are four quadratic resolvents, and the synthetic sequence indexes are 2, 3, 2, 2, so the general quartic equation can also be solved by roots. The galois group of the general equation of degree n is s(n), and the maximal normal subgroup of s(n) is A(n) (in fact, A(n) is a subgroup composed of even permutations in s(n)). If an arrangement can be expressed as the product of an even number of such arrangements, it is called an even arrangement. ), the number of elements of A(n) is half that of s(n), and the maximal normal subgroup of A(n) is the unit group I, so [s(n)/A(n)]=n! /(n！ /2)=2，[A(n)/I]=(n！ /2)/ 1=n！ /2,2 is a prime number, but when n ≥5, n! /2 is not a prime number, so general equations higher than quartic cannot be solved by roots. At this point, Galois completely solved the solvability problem of the equation.
By the way, Abel started with an exchange group. His starting point is different from Galois's, but their results are the same, all to prove that it is a solvable group. Galois also extended Abel equation and established an equation now called Galois equation. Each root of Galois equation is a rational function with two roots in the coefficient domain.
Four. The Historical Contribution of Galois Group Theory
Galois founded group theory to be applied to equation theory, but he was not limited to this, but extended the group theory and applied it to other research fields. It's a pity that Galois's group theory is too abstruse for people at the beginning of the 9th century to understand/kloc-0. Even mathematicians at that time could not understand the essence of his mathematical thought and work, so that his paper could not be published. More unfortunately, Galois 2 1 died young because of a stupid duel. We have to feel sorry for this genius. It was not until the 1960s that his theory was finally understood and accepted by people.
Galois group theory is recognized as one of the most outstanding mathematical achievements in19th century. He provided a comprehensive and thorough answer to the solvability of equations and solved the problem that puzzled mathematicians for hundreds of years. Galois group theory also gives a general method to judge whether geometric figures can be drawn with straightedge and compass, which satisfactorily solves the problem that bisecting any angle or cube is insoluble. Most importantly, group theory has opened up a brand-new research field, replaced calculation with structural research, changed the way of thinking from emphasizing calculation research to using structural concept research, and classified mathematical operations, which made group theory develop rapidly into a brand-new branch of mathematics and had a great influence on the formation and development of modern algebra. At the same time, this theory has a great influence on the development of physics and chemistry, and even on the emergence and development of structuralist philosophy in the twentieth century.
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Lu Youwen, editor. On ancient and modern mathematics. Tianjin: Tianjin Science and Technology Press, 1984.
A brief history of Chinese and foreign mathematics. A brief history of foreign mathematics. Shandong: Shandong Education Press, 1987.
Editor Wu Wenjun. Biographies of world famous scientists. Beijing: Science Press, 1994.
Tony Rothman, Biography of Galois, Chongqing Branch of Science and Technology Literature Publishing House, No.8, 1982, pp.8 1 ~ 92.