group

G0286200 (gro͞op)n.1. An assemblage of persons or objects gathered or located together; an aggregation: a group of dinner guests; a group of buildings near the road.2. A set of two or more figures that make up a unit or design, as in sculpture.3. A number of individuals or things considered or classed together because of similarities: a small group of supporters across the country.4. Linguistics A category of related languages that is less inclusive than a family.5. a. A military unit consisting of two or more battalions and a headquarters.b. A unit of two or more squadrons in the US Air Force, smaller than a wing.6. Chemistry a. Two or more atoms behaving or regarded as behaving as a single chemical unit.b. A column in the periodic table of the elements.7. Geology A stratigraphic unit, especially a unit consisting of two or more formations deposited during a single geologic era.8. Mathematics A set, together with a binary associative operation, such that the set is closed under the operation, the set contains an identity element for the operation, and each element of the set has an inverse element with respect to the operation. The integers form a group under the operation of ordinary addition.adj. Of, relating to, constituting, or being a member of a group: a group discussion; a group effort.v. grouped, group·ing, groups v.tr. To place or arrange in a group: grouped the children according to height.v.intr. To belong to or form a group: The soldiers began to group on the hillside.

[French groupe, from Italian gruppo, probably of Germanic origin.]Usage Note: Group as a collective noun can be followed by a singular or plural verb. It takes a singular verb when the persons or things that make up the group are considered collectively: The dance group is ready for rehearsal. Group takes a plural verb when the persons or things that constitute it are considered individually: The group were divided in their sympathies. See Usage Note at collective noun.

group

(ɡruːp) n1. a number of persons or things considered as a collective unit2. (Sociology) a. a number of persons bound together by common social standards, interests, etcb. (as modifier): group behaviour. 3. (Jazz) a small band of players or singers, esp of pop music4. (Biology) a number of animals or plants considered as a unit because of common characteristics, habits, etc5. (Grammar) grammar another word, esp in systemic grammar, for phrase¹6. (Accounting & Book-keeping) an association of companies under a single ownership and control, consisting of a holding company, subsidiary companies, and sometimes associated companies7. (Art Terms) two or more figures or objects forming a design or unit in a design, in a painting or sculpture8. (Military) a military formation comprising complementary arms and services, usually for a purpose: a brigade group. 9. (Military) an air force organization of higher level than a squadron10. (Chemistry) chem Also called: radical two or more atoms that are bound together in a molecule and behave as a single unit: a methyl group -CH3. Compare free radical11. (Chemistry) a vertical column of elements in the periodic table that all have similar electronic structures, properties, and valencies. Compare period⁸12. (Geological Science) geology any stratigraphical unit, esp the unit for two or more formations13. (Mathematics) maths a set that has an associated operation that combines any two members of the set to give another member and that also contains an identity element and an inverse for each element14. (Medicine) See blood groupvbto arrange or place (things, people, etc) in or into a group or (of things, etc) to form into a group[C17: from French groupe, of Germanic origin; compare Italian gruppo; see crop]

group

(grup)

n. 1. any collection or assemblage of persons or things; cluster; aggregation. 2. a number of persons or things ranged or considered together as being related in some way. 3. Also called radical. two or more atoms specifically arranged and usu. behaving as a single entity, as the hydroxyl group, –OH. 4. any of the vertical columns of elements in the periodic table. 5. a division of stratified rocks comprising two or more formations. 6. a. an administrative and tactical unit of the U.S. Army consisting of two or more battalions and a headquarters. b. an administrative and operational unit of the U.S. Air Force subordinate to a wing, usu. composed of two or more squadrons. 7. a section of an orchestra comprising the instruments of the same class. 8. an algebraic system that is closed under an associative operation, as multiplication or addition, and in which there is an identity element that, on operating on another element, leaves the second element unchanged, and in which each element has corresponding to it a unique element that, on operating on the first, results in the identity element. v.t. 9. to place together in a group, as with others. 10. to form into a group or groups. v.i. 11. to form a group. 12. to be part of a group. [1665–75; < French groupe < Italian gruppo « Germanic; akin to crop] usage: See collective noun.

group

(gro͞op)1. Two or more atoms bound together that act as a unit in a number of chemical compounds: a hydroxyl group.2. In the Periodic Table, a vertical column that contains elements having the same number of electrons in the outermost shell of their atoms. Elements in the same group have similar chemical properties. See Periodic Table.

group

1. A flexible administrative and tactical unit composed of either two or more battalions or two or more squadrons. The term also applies to combat support and combat service support units.
2. A number of ships and/or aircraft, normally a subdivision of a force, assigned for a specific purpose. Also called GP.

Group

a set of things collected as a unit. See also gathering.Examples: group of columns [three or four columns joined together on the same pedestal], 1731; of company, 1748; of crystals, 1830; of islands; of musicians; of partisans, 1809; of rocks, 1859; of singers; of trees; of woes, 1729; of words, 1748.

group

Past participle: grouped
Gerund: grouping

Imperative
group
group

Present
I group
you group
he/she/it groups
we group
you group
they group

Preterite
I grouped
you grouped
he/she/it grouped
we grouped
you grouped
they grouped

Present Continuous
I am grouping
you are grouping
he/she/it is grouping
we are grouping
you are grouping
they are grouping

Present Perfect
I have grouped
you have grouped
he/she/it has grouped
we have grouped
you have grouped
they have grouped

Past Continuous
I was grouping
you were grouping
he/she/it was grouping
we were grouping
you were grouping
they were grouping

Past Perfect
I had grouped
you had grouped
he/she/it had grouped
we had grouped
you had grouped
they had grouped

Future
I will group
you will group
he/she/it will group
we will group
you will group
they will group

Future Perfect
I will have grouped
you will have grouped
he/she/it will have grouped
we will have grouped
you will have grouped
they will have grouped

Future Continuous
I will be grouping
you will be grouping
he/she/it will be grouping
we will be grouping
you will be grouping
they will be grouping

Present Perfect Continuous
I have been grouping
you have been grouping
he/she/it has been grouping
we have been grouping
you have been grouping
they have been grouping

Future Perfect Continuous
I will have been grouping
you will have been grouping
he/she/it will have been grouping
we will have been grouping
you will have been grouping
they will have been grouping

Past Perfect Continuous
I had been grouping
you had been grouping
he/she/it had been grouping
we had been grouping
you had been grouping
they had been grouping

Conditional
I would group
you would group
he/she/it would group
we would group
you would group
they would group

Past Conditional
I would have grouped
you would have grouped
he/she/it would have grouped
we would have grouped
you would have grouped
they would have grouped

Thesaurus

Noun	1.	group - any number of entities (members) considered as a unitgroupingabstract entity, abstraction - a general concept formed by extracting common features from specific exampleshuman beings, human race, humankind, humans, mankind, humanity, world, man - all of the living human inhabitants of the earth; "all the world loves a lover"; "she always used `humankind' because `mankind' seemed to slight the women"arrangement - an orderly grouping (of things or persons) considered as a unit; the result of arranging; "a flower arrangement"straggle - a wandering or disorderly grouping (of things or persons); "a straggle of outbuildings"; "a straggle of followers"kingdom - a basic group of natural objectsbiological group - a group of plants or animalsbiotic community, community - (ecology) a group of interdependent organisms inhabiting the same region and interacting with each otherpeople - (plural) any group of human beings (men or women or children) collectively; "old people"; "there were at least 200 people in the audience"social group - people sharing some social relationaggregation, collection, accumulation, assemblage - several things grouped together or considered as a wholeedition - all of the identical copies of something offered to the public at the same time; "the first edition appeared in 1920"; "it was too late for the morning edition"; "they issued a limited edition of Bach recordings"electron shell - a grouping of electrons surrounding the nucleus of an atom; "the chemical properties of an atom are determined by the outermost electron shell"ethnic group, ethnos - people of the same race or nationality who share a distinctive culturerace - people who are believed to belong to the same genetic stock; "some biologists doubt that there are important genetic differences between races of human beings"association - (ecology) a group of organisms (plants and animals) that live together in a certain geographical region and constitute a community with a few dominant speciesswarm, cloud - a group of many things in the air or on the ground; "a swarm of insects obscured the light"; "clouds of blossoms"; "it discharged a cloud of spores"subgroup - a distinct and often subordinate group within a groupsainthood - saints collectivelycitizenry, people - the body of citizens of a state or country; "the Spanish people"population - a group of organisms of the same species inhabiting a given area; "they hired hunters to keep down the deer population"hoi polloi, masses, the great unwashed, multitude, people, mass - the common people generally; "separate the warriors from the mass"; "power to the people"varna - (Hinduism) the name for the original social division of Vedic people into four groups (which are subdivided into thousands of jatis)circuit - (law) a judicial division of a state or the United States (so-called because originally judges traveled and held court in different locations); one of the twelve groups of states in the United States that is covered by a particular circuit court of appealssystem, scheme - a group of independent but interrelated elements comprising a unified whole; "a vast system of production and distribution and consumption keep the country going"series - a group of postage stamps having a common theme or a group of coins or currency selected as a group for study or collection; "the Post Office issued a series commemorating famous American entertainers"; "his coin collection included the complete series of Indian-head pennies"actinide, actinoid, actinon - any of a series of radioactive elements with atomic numbers 89 through 103lanthanide, lanthanoid, lanthanon, rare earth, rare-earth element - any element of the lanthanide series (atomic numbers 57 through 71)halogen - any of five related nonmetallic elements (fluorine or chlorine or bromine or iodine or astatine) that are all monovalent and readily form negative ions
	2.	group - (chemistry) two or more atoms bound together as a single unit and forming part of a moleculechemical group, radicalchemical science, chemistry - the science of matter; the branch of the natural sciences dealing with the composition of substances and their properties and reactionsbuilding block, unit - a single undivided natural thing occurring in the composition of something else; "units of nucleic acids"acyl, acyl group - any group or radical of the form RCO- where R is an organic group; "an example of the acyl group is the acetyl group"alcohol group, alcohol radical - the chemical group -OHaldehyde group, aldehyde radical - the chemical group -CHOalkyl, alkyl group, alkyl radical - any of a series of univalent groups of the general formula CnH2n+1 derived from aliphatic hydrocarbonsallyl, allyl group, allyl radical - the univalent unsaturated organic radical C3H5; derived from propyleneamino, amino group - the radical -NH2amyl - a hydrocarbon radical that occurs in many organic compoundsazido group, azido radical - the univalent group N3- derived from hydrazoic acidazo group, azo radical - the bivalent group -N=N- united to two hydrocarbon groupsbenzyl, benzyl group, benzyl radical - the univalent radical derived from toluenebenzoyl group, benzoyl radical - the univalent radical derived from benzoic acidmolecule - (physics and chemistry) the simplest structural unit of an element or compoundarsenic group, cacodyl group, cacodyl radical, cacodyl - the univalent group derived from arsinecarbonyl group - the bivalent radical COcarboxyl, carboxyl group - the univalent radical -COOH; present in and characteristic of organic acidschromophore - the chemical group that gives color to a moleculecyanide group, cyanide radical, cyano group, cyano radical - the monovalent group -CN in a chemical compoundglyceryl - a trivalent radical derived from glycerol by removing the three hydroxyl radicalshydrazo group, hydrazo radical - the bivalent group -HNNH- derived from hydrazinehydroxyl, hydroxyl group, hydroxyl radical - the monovalent group -OH in such compounds as bases and some acids and alcoholsketone group - a group having the characteristic properties of ketonesmethylene, methylene group, methylene radical - the bivalent radical CH2 derived from methanepropyl, propyl group, propyl radical - the monovalent organic group C3H7- obtained from propanebutyl - a hydrocarbon radical (C4H9)nitro group - the group -NO3nitrite - the radical -NO2 or any compound containing it (such as a salt or ester of nitrous acid)uranyl, uranyl group, uranyl radical - the bivalent radical UO2 which forms salts with acidsvinyl, vinyl group, vinyl radical - a univalent chemical radical derived from ethylene
	3.	group - a set that is closed, associative, has an identity element and every element has an inversemathematical groupsubgroup - (mathematics) a subset (that is not empty) of a mathematical groupAbelian group, commutative group - a group that satisfies the commutative lawset - (mathematics) an abstract collection of numbers or symbols; "the set of prime numbers is infinite"
Verb	1.	group - arrange into a group or groups; "Can you group these shapes together?"class, classify, sort out, assort, sort, separate - arrange or order by classes or categories; "How would you classify these pottery shards--are they prehistoric?"regroup - reorganize into new groupsbracket - classify or groupchunk, collocate, lump - group or chunk together in a certain order or place side by sidebatch - batch together; assemble or process as a batch
	2.	group - form a group or group together aggroupteam, team up - form a team; "We teamed up for this new project"embed - attach to, as a journalist to a military unit when reporting on a war; "The young reporter was embedded with the Third Division"gang, gang up - act as an organized grouppool - join or form a pool of peoplebrigade - form or unite into a brigadeforegather, forgather, gather, assemble, meet - collect in one place; "We assembled in the church basement"; "Let's gather in the dining room"

group

noun1. crowd, company, party, band, troop, pack, gathering, gang, bunch, congregation, posse (slang), bevy, assemblage The trouble involved a small group of football supporters.2. organization, body, association, league, circle Members of an environmental group are staging a protest inside a chemical plant.3. faction, set, camp, clique, coterie, schism, cabal a radical group within the Communist Party4. category, class, section, grouping, order, sort, type, division, rank, grade, classification The recipes are divided into groups according to their main ingredients.5. band, ensemble, combo ELP were the progressive rock group par excellence.6. cluster, collection, formation, clump, aggregation a small group of islands off northern Japanverb1. arrange, order, sort, class, range, gather, organize, assemble, put together, classify, dispose, marshal, bracket, assort The fact sheets are grouped into seven sections.2. unite, associate, gather, cluster, get together, congregate, band together We want to encourage them to group together as one big purchaser.

group

noun1. A number of individuals making up or considered a unit:array, band, batch, bevy, body, bunch, bundle, clump, cluster, clutch, collection, knot, lot, party, set.2. A number of persons who have come or been gathered together:assemblage, assembly, body, company, conclave, conference, congregation, congress, convention, convocation, crowd, gathering, meeting, muster, troop.Informal: get-together.3. A group of people sharing an interest, activity, or achievement:circle, crowd, set.verb1. To bring together:assemble, call, cluster, collect, congregate, convene, convoke, gather, get together, muster, round up, summon.2. To come together:assemble, cluster, collect, congregate, convene, forgather, gather, get together, muster.3. To distribute into groups according to kinds:assort, categorize, class, classify, pigeonhole, separate, sort (out).4. To assign to a class or classes:categorize, class, classify, distribute, grade, pigeonhole, place, range, rank, rate.

Translations

小组批组群聚集

group

(gruːp) noun1. a number of persons or things together. a group of boys.群2. a group of people who play or sing together. a pop group; a folk group.小组 verb to form into a group or groups. The children grouped round the teacher.聚集

group

→ 批zhCN, 组zhCN

My blood group is O positive → 我是O型血

group

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.

Bibliography

See R. P. Burn, Groups (1987); J. A. Green, Sets and Groups (1988).

group

Any 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).

Group

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

Group

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.

Figure 1

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 . . . i_n

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.

REFERENCES

Aleksandrov, 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

Group

(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.

group

[grüp] (astronomy) A number of stars moving in the same direction with the same speed. (chemistry) A family of elements with similar chemical properties. A combination of bonded atoms that behave as a unit under certain conditions, for example, the sulfate group, SO₄^2-. (communications) A communication transmission subdivision containing a number of voice channels, either within a supergroup or separately, normally comprised of up to 12 voice channels occupying the frequency band 60-108 kilohertz; each voice channel may be multiplexed for teletypewriter operation, if required; the number of voice channels which may be simultaneously multiplexed for teletypewriter operation will vary according to equipment design. (geology) A lithostratigraphic material unit comprising several formations. (mathematics) A set G with an associative binary operation where g₁· g₂ always exists and is an element of G, each g has an inverse element g ^-1, and G contains an identity element.

group

1. a small band of players or singers, esp of pop music 2. a number of animals or plants considered as a unit because of common characteristics, habits, etc. 3. two or more figures or objects forming a design or unit in a design, in a painting or sculpture 4. Chem two or more atoms that are bound together in a molecule and behave as a single unit 5. a vertical column of elements in the periodic table that all have similar electronic structures, properties, and valencies 6. Geology any stratigraphical unit, esp the unit for two or more formations 7. Maths a set that has an associated operation that combines any two members of the set to give another member and that also contains an identity element and an inverse for each element 8. See blood group

group

A group G is a non-empty set upon which a binary operator* is defined with the following properties for all a,b,c in G:

Closure: G is closed under *, a*b in GAssociative: * is associative on G, (a*b)*c = a*(b*c)Identity: There is an identity element e such thata*e = e*a = a.Inverse: Every element has a unique inverse a' such thata * a' = a' * a = e. The inverse is usuallywritten with a superscript -1.

See group

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group

a committee is a group of men who keep minutes and waste hours

ginger group

call (a group) together

the hearts and minds of (the members of some group)

nose out

splinter group

group text

polarize (one group of people) into (two groups of people)

group (someone or something) around (someone or something)

group (someone or something) together

group under (something)

group-grope

group someone or something around someone or something

group someone or something together

group something under something

nose out (of something)

nose someone or a group out

nose out

splinter group

ginger group

nose out

group-grope

group

group,

Bibliography

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Group

REFERENCES

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GROUP

group

Synonyms for group

noun crowd

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noun organization

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noun faction

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noun category

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noun band

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noun cluster

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verb arrange

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verb unite

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Synonyms for group

noun a number of individuals making up or considered a unit

Synonyms

noun a number of persons who have come or been gathered together

Synonyms

noun a group of people sharing an interest, activity, or achievement

Synonyms

verb to bring together

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verb to come together

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verb to distribute into groups according to kinds

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