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group,in mathematics, system consisting of a set of elements and a binary operation a+b defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e., if a and b are elements of the set, then the element that results from combining a and b under the operation is also an element of the set; (2) the operation satisfies the associative lawassociative law,
in mathematics, law holding that for a given operation combining three quantities, two at a time, the initial pairing is arbitrary; e.g., using the operation of addition, the numbers 2, 3, and 4 may be combined (2+3)+4=5+4=9 or 2+(3+4)=2+7=9.
..... Click the link for more information. ; i.e., a+(b+c)=(a+b)+c, where + represents the operation and a, b, and c are any three elements; (3) there exists an identity element I in the set such that a+I=a for any element a in the set; (4) there exists an inverse a−1 in the set for every a such that a+a−1=I. If, in addition to satisfying these four axioms, the group also satisfies the commutative lawcommutative law,
in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or as 5+2=7.
..... Click the link for more information. for the operation, i.e., a+b=b+a, then it is called a commutative, or Abelian, group. The real numbers (see numbernumber,
entity describing the magnitude or position of a mathematical object or extensions of these concepts. The Natural Numbers
Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their
..... Click the link for more information. ) form a commutative group both under addition, with 0 as identity element and −a as inverse, and, excluding 0, under multiplication, with 1 as identity element and 1/a as inverse. The elements of a group need not be numbers; they may often be transformations, or mappings, of one set of objects into another. For example, the set of all permutations of a finite collection of objects constitutes a group. Group theory has wide applications in mathematics, including number theory, geometry, and statistics, and is also important in other branches of science, e.g., elementary particle theory and crystallography.
See R. P. Burn, Groups (1987); J. A. Green, Sets and Groups (1988).
groupAny collectivity or plurality of individuals (people or things) bounded by informal or formal criteria of membership. A social group exists when members engage in social interactions involving reciprocal ROLES and integrative ties. The contrast can be drawn between a social group and a mere social category, the latter referring to any category of individuals sharing a socially relevant characteristic (e.g. age or sex), but not associated within any bounded pattern of interactions or integrative ties. In terms of membership, social groups may be either relatively open and fluid (e.g. friendship groups), or closed and fixed (e.g. Masonic Lodges).
Any social group, therefore, will have a specified basis of social interaction, though the nature and extent of this will vary greatly between groups. Social groups of various types can be seen as the building blocks from which other types and levels of social organization are built. Alternatively, the term ‘social group’, as for Albion SMALL (1905), is ‘the most general and colourless term used in sociology to refer to combinations of persons’. See also PRIMARY GROUP, GROUP DYNAMICS, REFERENCE GROUP, SOCIAL INTEGRATION AND SYSTEM INTEGRATION, SOCIETY, DESCENT GROUP, PEER GROUP, PRESSURE GROUP, STATUS GROUP, IN-GROUP AND OUT-GROUP.
The best-known theoretical and experimental approach in the study of group dynamics, and the one with which the term is most associated, is the FIELD THEORY of Kurt Lewin (1951); however, an awareness of the importance of group dynamics in a more general sense is evident in the work of many sociologists and social psychologists, including SIMMEL, MAYO, MORENO, Robert Bales (1950) and PARSONS (see DYAD AND TRIAD, HAWTHORN EFFECT, SOCIOMETRY, OPINION LEADERS AND OPINION LEADERSHIP, CONFORMITY, GROUP THERAPY).
in geology, a subdivision of the general strati-graphic scale comprising all rocks formed during a single geological era.
The term “group” was adopted at the second session of the International Geological Congress in 1881. Taking issue with this decision, American geologists use the term “erathema” in place of “group,” using the latter to mean a subdivision of a local stratigraphie scale. Groups are subdivided into systems; several groups constitute an eono-thema. Every group corresponds to a certain stage in the development of the earth and the earth’s crust and is characterized by unique geological deposits and fossils. There are five groups: the Archean. the Proterozoic, the Paleozoic, the Mesozoic. and the Cenozoic.
B. M. KELLER
one of the fundamental concepts of modern mathematics. The theory of groups studies the properties of operations that occur most frequently in mathematics and its applications from the most general point of view (examples of such operations are multiplication of numbers, addition of vectors, successive accomplishment of transformations, and so on). The generality of the theory of groups, and thus its wide applicability, stems from the fact that the theory studies the properties of operations in their pure form, ignoring the nature of the particular operation as well as the nature of the elements operated on. At the same time, it must be stressed that the theory of groups is restricted to the study of operations that satisfy a number of properties that are part of the definition of a group (see below).
One way of arriving at the concept of a group is to study the symmetries of geometric figures. Thus, a square (Figure 1,a) is a symmetric figure in the sense that, say, a clockwise rotation ø of the square about its center or a reflection ∊ about its diagonal AC maps the square to itself. There are eight different motions that map the square to itself. In the case of the circle (Figure 1,b) the number of such motions is infinite; each rotation of the circle about its center is a motion of the required type. In the case of the figure represented in Figure 1, c the only motion that maps it to itself is the identity transformation, that is, the transformation which leaves each point of the figure fixed.
The set G of motions that map a figure to itself is a measure of its symmetry; the larger the set G the more symmetric the figure. We define by means of the set G the composition, that is, an operation of the elements of G in the following manner: If ø, ∊ are two motions in G, then by their composition, or product Ѱ and ∊, we mean the motion øO∊ that is the result of applying to the figure first the motion Ѱ and then the motion ∊. For example, if ø and ∊ are the abovementioned motions of a square, then øO∊ is the reflection of the square in the line passing through the midpoints of its sides AB and CD. The set G of motions of a figure together with the composition defined on G is called the group of symmetries of the figure. It is clear that the composition of the elements of G has the following properties: (l) For all ø, Ѱ, θ in G(øO∊) = øO(∊O θ); (2) G contains an element ∊ such that ∊O ø = øO∊ = ø for all ø in G; (3) for each ø in G there is an element ø-1 in G such that øO ø-1 = ø-OѰ = ∊. In fact, the role of ø is played by the identity transformation and the role of Ѱ-1 is played by the inverse of Ѱ, that is, Ѱ-1 is the transformation that reverses the effect of Ѱ on the points of the figure.
The general (formal) definition of a group is as follows: A set G of arbitrary elements is called a group if there is defined on it an operation O that associates with every two of its elements ø,O Ѱ a unique element øOѰ in G such that properties (1), (2), and (3) hold.
For example, if G is the set of integers and the operation is ordinary addition, then G is a group; here, ∊ is the number 0 and ø-1 is the number -ø. The subset H of even numbers is itself a group under addition. We say that H is a subgroup of G. We note that either of these groups satisfies the additional requirement: (4) øOѰ = ѰOø for all ø,Ѱ in the group. A group with this additional property is said to be commutative, or Abelian.
Another example of a group. By a permutation of the symbols 1, 2, . . . , n we mean the array
1, 2,..., n
i1i2 . . . in
where the symbols in the lower row are a rearrangement of the symbols 1, 2, . . . , n. The product of two permutations ø,Ѱ, is defined as follows: if in ø the symbol below x is y and in Ѱ the symbol below y is z, then in øOѰ we put z below x. For example,
It can be shown that the set of permutations of η symbols under the above operation forms a group. For n ≥ 3 this group is non-Abelian.
History. The concept of a group served in many respects as a model for the restructuring of algebra and, more generally, of mathematics that took place at the turn of the 20th century. The sources of the concept of a group are evident in several disciplines, the most important of which is the theory of solvability of algebraic equations by radicals. In 1711 the French mathematicians J. Lagrange and A. Vandermonde were the first to apply permutations to the theory of algebraic equations (Lagrange’s paper On the Solution of Algebraic Equations is especially important in the development of the theory of groups). P. Ruffini published a number of papers (in 1799 and later) devoted to the demonstration of the unsolva-bility of the general quintic by radicals in which he made systematic use of the closure of permutations under multiplication and determined in essence the subgroups of the group of permutations on five symbols. The deep ties between properties of groups of permutations and properties of equations were demonstrated by the Norwegian mathematician N. Abel (1824) and the French mathematician E. Galois (1830). It was Galois who must be credited with such concrete advances in group theory as the elucidation of the role of normal subgroups in the problem of solvability of equations by radicals, and the proof of the simplicity of the alternating group of order n ≥ 5. While Galois introduced the term “group” (le group), he did not define it rigorously. A treatise on groups of permutations published in 1870 by the French mathematician C. Jordan played an important role in the sys-tematization and development of group theory.
The idea of a group emerged independently and for other reasons in geometry in the middle of the 19th century when the single classical geometry was replaced by numerous “geometries” and mathematicians were faced with the urgent problem of clarifying the connections and the relations between them. A resolution of this problem came from investigations in the area of projective geometry concerned with the behavior of figures under various transformations. Gradually, interest in these studies shifted to the transformations themselves and to their classification. The German mathematician A. Möbius devoted a great deal of attention to such a “study of geometric relations.” The final stage in these developments was the Erlanger Programm of the German mathematician F. Klein (1872), which used groups of transformation as the basis for the classification of geometries. Specifically, a geometry is defined by a group of transformations of space, and the only properties of figures that belong to the geometty in question are the properties invariant with respect to the transformations in this group.
The third source of the group concept is number theory. Already in 1761, L. Euler in his study “Residues Resulting From the Division of Powers” used, in essence, consequences and residue classes. In group-theoretical terms this amounts to decomposition of a group into cosets with respect to a subgroup. K. Gauss, in his Disquisitiones Arithmeticae (1801), studied, among others, the cyclotomic equation and, in essence, determined the subgroups of its Galois group. In the same work Gauss studied “composition of binary quadratic forms” and showed, in effect, that, relative to this composition, the equivalence classes of forms constituted a finite Abelian group. The German mathematician L. Kronecker developed these ideas (1870) and came close to the discovery of the fundamental theorem on Abelian groups, although he did not state it explicitly.
Until the end of the 19th century group-theoretical forms of thought existed independently in various areas of mathematics. The recognition of their essential unity led to the formulation of the modern abstract concept of a group. Some of the prominent mathematicians connected with this development were the Norwegian mathematician S. Lie and the German mathematician F. Frobenius. Thus, as early as 1895, Lie defined a group as a set of transformations closed under the rule of composition and satisfying conditions (1), (2), and (3). After the publication (in 1916) of O. Iu. Shmidt’s work Abstract Theory of Groups, the study of groups without the assumption of finiteness and without any assumptions about the nature of the group elements became an independent branch of mathematics.
Theory of groups. The ultimate goal of the theory of groups is the classification of all possible group compositions. There are a number of branches of group theory. These branches differ from one another by the additional conditions imposed on the group operation or by the introduction into the group of additional structures related in a definite way to the group operation. The following are the major branches of group theory.
(1) The theory of finite groups. The fundamental problem of this oldest branch of the theory of groups is the classification of so-called finite simple groups, which are the building blocks for the construction of arbitrary finite groups. One of the profoundest results in this theory is the theorem which asserts that every non-Abelian simple finite group has an even number of elements.
(2) Theory of Abelian groups. The starting point of many investigations in this area is the fundamental theorem on finitely generated Abelian groups, which elucidates their structure completely.
(3) Theory of solvable and nilpotent groups. The concept of a solvable group is a generalization of the concept of an Abelian group. In essence it goes back to Galois and is closely connected with the problem of solvability of equations by radicals.
For finite groups these concepts can be defined in many equivalent ways that cease being equivalent for infinite groups. This leads to the study of so-called generalized solvable groups and generalized nilpotent groups.
(4) The theory of groups of transformations. Initially mathematicians worked with groups of transformations rather than with abstract groups. Notwithstanding the evolution of the latter concept, the theory of groups of transformations remained an important part of the general theory. A typical problem in this theory is “What are the abstract properties of a group defined as a group of transformations of a given set?” Of special interest are groups of permutations and groups of matrices.
(5) The theory of representations of groups is an important tool for the study of abstract groups. Representation of an abstract group by means of a concrete group (say, as a group of permutations or a group of matrices) makes it possible to carry out sophisticated calculations and to establish with their aid important abstract properties. The theory of group representations has been especially important in the theory of finite groups where its application has yielded a number of results that are as yet inaccessible to abstract methods.
(6) It is possible to introduce into a group additional structures that dovetail with the group operation. A relevant example is the concept of a topological group (in such groups, the group operation is in a certain sense continuous). The oldest part of the theory of topological groups is the theory of Lie groups.
The theory of groups is one of the most developed fields of algebra and has numerous applications in mathematics as well as outside mathematics. For example, E. S. Fedorov (1890) used group theory to solve the problem of the classification of regular systems of points in space, which is one of the main problems of crystallography. This was the first direct application of group theory to a problem in the natural sciences. The theory of groups plays a major role in physics, for example, in quantum mechanics, where the concepts of symmetry and representation of groups by linear transformations are widely used.
REFERENCESAleksandrov, P. S. Vvedenie v teoriiu grupp, 2nd ed. Moscow, 1951.
Mal’tsev, A. I. “Gruppy i drugie algebraicheskie sistemy.” In the book Matematika, ee soderzhanie, melody i znachenie, vol. 3. Moscow, 1956. Pages 248–331.
Kurosh, A. G. Teoriia grupp, 3rd ed. Moscow, 1967.
Hall, M. Theory of Groups. Moscow, 1962. (Translated from English.)
Van der Waerden, B. L. Metod teorii grupp v kvantovoi mekhanike. Kharkov, 1938. (Translated from German.)
Shmidt, O. Iu. “Abstraktnaia teoriia grupp.” In his book Izbr. trudy: Matematika. Moscow, 1959.
Fedorov, E. S. “Simmetriia pravil’nykh sistem figur.” In his book Simmetriia i struktura kristallov: Osnovnye raboty. Moscow, 1949.
Wussing, H. Die Genesis des abstraken Gruppenbe griffes. Berlin, 1969. Pages 1–258.
M. I. KARGAPOLOV and
IU. I. MERZLIAKOV
(1) The unification of units under the general command of a senior commander for the performance of an operational (combat) mission. In the Soviet armed forces during the Great Patriotic War (1941–45) operational groups were formed which carried out missions in frontline offensive and defensive operations, usually separated from the main forces; mobile groups were used to develop the offensive in the depth of the enemy defense after breaking through it. Artillery (mortar) and antiaircraft artillery groups were formed to support combat action.
(2) During the 1930’s a part of the battle formation of large units of the Soviet ground forces; the formation was divided into assault, holding, and fire groups.
(3)Regular organization in the US armed forces—the army aviation group and the special forces group (for carrying out diversionary and subversive activity on enemy territory); also in the armed forces of Great Britain—the infantry brigade group, which is the primary large combined arms tactical unit.
Closure: G is closed under *, a*b in G Associative: * is associative on G, (a*b)*c = a*(b*c) Identity: There is an identity element e such that a*e = e*a = a. Inverse: Every element has a unique inverse a' such that a * a' = a' * a = e. The inverse is usually written with a superscript -1.