Firstly, an arrangement group consisting of some arrangements of n characters is generated. It was introduced and developed by J.-L. Lagrange, P. Ruffini, N.H. Abel and E. Galois when they were studying the central problem of algebra at that time, that is, whether a polynomial equation with more than five degrees can be solved by roots, and it was used to solve this central problem effectively. The permutation group formed by some permutations between the roots of the polynomial equation of degree n in the number domain is defined as the Galois group of the equation. In 1832, Galois proved that a polynomial equation of degree n with one variable can be solved by roots if and only if the Galois group of the equation is a "solvable group" (see finite group). Because the Galois group of a general univariate equation of degree n is a symmetric group Sn with n characters, and Sn is not a solvable group when n≥5, a univariate equation of degree five or more cannot be solved by roots. Galois also introduced some important concepts such as isomorphism and normal subgroups of permutation groups. It should be pointed out that A.-L. Cauchy published the first paper on permutation groups as early as 18 15, and did a lot of work on permutation groups from 1844 to 1846. Regarding the systematic knowledge of permutation groups and Galois's application in the study of equation theory, Galois's manuscript was written on the eve of his death in a duel, and it was not until later that it was well introduced and further developed in C. Jordan's masterpiece Monographs on Permutation and Algebraic Equations. Permutation group is the first and most important source that ultimately produces and forms abstract groups.
In number theory, Lagrange and C.F. Gauss studied quadratic form with the same discriminant D, that is, f = ax 2+2bxy+cy 2, where a, b and с are integers, x and y are integer values, and d = b 2-a с is a constant value. J.W.R Dai Dejin introduced finite commutative groups or even finite groups in the study of algebraic number theory in 1858 and L.Kroneck in 1870. These are the second main sources that lead to the emergence of abstract group theory.
Under the influence of Jordan's monograph, (C.) F. Klein pointed out in his famous Erlangen program 1872 that the classification of geometry can be carried out through infinite continuous transformation groups. Klein and (J.-) H. Poincare have used other types of infinite groups (that is, discrete groups or discontinuous groups) in the study of "automorphism functions". Around 1870, Sophie Lee began to study continuous transformation groups, that is, analytic transformation Lie groups, to clarify the solutions of differential equations and classify them. This infinite transformation group theory has become the third main source leading to the emergence of abstract group theory.
A.Gloria has already mentioned the concept of close to finite abstract groups in her papers 1849, 1854 and 1878. The work of F.G. Frobenius in 1879, E.Neto in 1882, and W.F.A.von Vondick in 1882 ~ 1883 also promoted this understanding. In the 1980s of 19, mathematicians finally successfully summed up the axiomatic system of abstract group theory, which was recognized in about 1890. At the beginning of the 20th century, E.V. Huntington, E.H. Moore and L.E. Dixon all gave various independent axiomatic systems of abstract groups, which are consistent with modern definitions.
During the period of1896 ~1911,W. burnside's Theory of Finite Groups was published twice, and gained a lot. G. Frobenius, W. burnside and I. Schur established the matrix representation theory of finite groups, and the finite group theory has been formed. Infinite group theory also had monographs in the early 20th century, such as1916 οю Schmidt's works. The development of group theory led to the rise of abstract algebra in 1930s. Especially in the last 30 years, finite group theory has made great progress. At the beginning of 198 1, the complete solution of the classification problem of finite simple groups is an outstanding achievement. At the same time, infinite group theory has also made rapid progress.
Today, the concept of group has been generally regarded as one of the most basic concepts in mathematics and many applications. It not only plays an important role in many branches of mathematics such as geometry, algebraic topology, function theory and functional analysis, but also forms some new disciplines such as topological groups, lie groups, algebraic groups and arithmetic groups. They also have other structures related to group structure, such as topology, analytic manifold, algebraic cluster and so on. , and used in crystallography, theoretical physics, quantum chemistry, even (algebraic) coding, automata theory, etc. As the product of popularizing the concept of "group", semigroup and simple semigroup theory and their applications in computer science and operator theory have also made great progress. The research on computer methods and programs of group theory has developed rapidly.
As far as scientific content is concerned, group theory belongs to the category of mathematics and is applied in many branches of mathematics. It is also widely used in physics, chemistry, engineering science and many other fields, especially physics. From the study of symmetry and conservation law in classical physics, to the exploration of angular momentum theory and dynamic symmetry in quantum mechanics, to the application of isospin, overload and SU(3) symmetry in modern elementary particle physics, the brilliance of group theory shines brightly. Generally speaking, we often use group theory to study symmetry, which can reflect the properties of some variables under certain changes. It is also related to physical equations. Lie groups, which are often mentioned in basic physics, are similar to Galois groups used to solve algebraic equations and are closely related to the solutions of differential equations.
Physically, permutation group is a very important group. Permutation groups include S3 group, two-dimensional rotation group, three-dimensional rotation group and Lorenz group corresponding to four-dimensional space-time. Lorenz group plus four-dimensional transformation constitutes poincare group.
In addition, the early questions about crystal structures in crystallography have also been answered through the study of Fedoroff groups in group theory. According to group theory, there are only 230 different crystal structures in space.
When studying groups, it is more convenient to replace group elements with representations, because group elements are generally abstract things. Representation can be regarded as a matrix, which has the same properties as group elements. Irreducible representation and unit representation are important concepts in representation theory.
In the eyes of many mathematicians who study group theory, that is, abstract group theory, mathematicians care about the operational relationship between elements, that is, the structure of a group, without considering the specific meaning of the elements of the group. For example, according to Kelley theorem, any group is isomorphic to a permutation group composed of elements of the group. Therefore, especially for studying finite groups, it is an important problem to study permutation groups.