Read and learn the basic vocabulary terms.
| concern (n) | [kqn¢sq:n] | интерес; отношение; забота |
| contemporary (a) | [kqn¢tempqrqrI] | современный |
| evolve (v) | [I¢vOlv] | развиваться, эволюционировать |
| consequence (n) | [¢kOnsIkwqns] | значение, важность; следствие |
| abstract (v) | [xb¢strxkt] | рассматривать отвлеченно, абстрагировать; отделять |
| curious (a) | [¢kjuqrIqs] | удивительный, странный |
| awareness (n) | [q¢wEqnIs] | осведомленность, информированность |
| explicitly (ad) | [Iks¢plIsItlI] | явно, в явном виде |
| implicitly (ad) | [Im¢plIsItlI] | неявно, в неявном виде (форме) |
| entity (n) | [¢entItI] | объект, данность; сущность; нечто объективно существующее |
| unique (a) | [ju¢nI:k] | единственный, однозначный; особый, исключительный |
| closure (n) | [¢klouZq] | замыкание |
| associative (a) | [q¢souSIqtIv] | ассоциативный; сочетательный |
| identity (a) | [aI¢dentItI] | тождество; единица, единичный элемент |
| inverse (n) | [¢In¢vq:s] | обратная величина (функция, оператор); инверсия, обращение |
| fascination (n) | ["fxsI¢neISqn] | волшебная сила; очарование |
| infinite (a) | [¢InfInIt] | бесконечный, бесконечно большой; неограниченный |
| finite(a) | [¢faInaIt] | конечный, ограниченный |
| select (v) | [sI¢lekt] | выбирать, отбирать |
| verify (v) | [¢verIfaI] | проверять, сличать |
| minor (a) | [¢maInq] | малый; незначительный |
| property (n) | [¢prOpqtI] | свойство, качество |
| sense (v) | [¢sens] | осознавать; понимать; чувствовать |
| structure (n) | [¢strAktSq] | структура; форма, вид; система |
Memorise the following word combinations
| the distributive law | распределительный (дистрибутивный) закон |
| to combine together | сочетать, объединять, комбинировать |
| the law of combination | закон комбинаций (сочетания) |
| a scalar product | скалярное произведение |
| a progression of abstractions | последовательность абстракций |
| to belong to the group | принадлежать группе |
| to make no difference | не иметь значения |
| i.e. (id est) that is | то есть |
Notes to be paid attention to
| I have not asked for help, neither do I desire it. | Помощи я не прошу и в помощи не нуждаюсь |
| The first attempt was not successful and neither was the second. | Первая попытка была неудачной, вторая также. |
| a matter of great consequence | дело большой важности |
| It is of no consequence. | Это неважно. Это не имеет значения. |
| It follows as a logical consequence that … | Логическим выводом из этого является то, что … Отсюда следует, что … |
| Time and space are entities | Время и пространство реально существуют |
| an inverse of number | обратная величина числа |
| an inverse of point | инверсия точки |
| of minor interest | не представляющий большого интереса |
TEXT A
GROUP THEORY
The theory of groups, a central concern of contemporary maths, has evolved through a progression of abstractions. A group is one of the simplest and the most important algebraic structures of consequence.
Some of the components of the group concept (i.e., those essential properties that were later abstracted and formulated as axioms) were recognized as early as 1650 B.C. when the Egyptians showed a curious awareness that something was involved in assuming that ab=ba. The Egyptians also freely used the distributive law, namely, a(b+c)=ab+ac.
The group concept was not recognized as explicitly as were some of its axioms, but even so it was implicitly sensed and used before Abel and Galois brought it into focus and before Cayley (1854) defined a general abstract group.
A group in its most abstract form consists of a number of entities known as the elements of a group which can be combined together according tovarious axioms. The form of combination is one in which two elements combine together to give a unique third element. This generalizes the familar operations of addition in which two numbers are added toform a third number or of multiplication in which they are multiplied to give their product. For convenience, this abstract operation is called multiplication and denoted by writing the elements close together. The order in which the elements are written is usually important.
The general definition of a group. A collection of elements, G, will be called a groupif its elements А, В, С... can be combined together (multiplied) in a way which satisfies the four axioms:
I. Closure.The product of any two elements of the group is a unique element which also belongs to the group.
II. Associative. When three or more elements are multiplied, the order of the multiplications makes no difference, i. e.
A(BC) = (AB)C=ABC.
III. Identity. Among the elements there is an identity element denoted
by I, with the property of leaving the elements unchanged on multiplication, i. e.
AI = IA = A.
IV. Inverses. Each element, A, in the group has an inverse (or reciprocal)
A-1such that AA-1=A-1 A = I.
A group does not need to have an infinite number of elements, the four elements 1, i, -1, -iform a finite group under multiplication. Neither does it need to use ordinary multiplication as the formcombination. The positive and negative integers form a group with addition as the law of combination, the number 0 as the identity and a change of sign to denote the inverse.
Subgroups. When a number of elements selected from a group do themselves form a group it is known as a subgroup. The identity element for the group is also the identity element for the subgroup. The closure and existence of inverses are, therefore, the only laws that need to be verified individually. Every group has at least one subgroup, namely, the minor one consisting of the identity element alone.
Post-Reading Activity
Ex. 8. Answer the following questions.
1. What is the central concern of contemporary maths? 2. Where can we find the traces of the components of the group concept? 3. Who was the first to bring the notion of a group concept info focus? 4. When and by whom was the definition of a general abstract group given? 5. What is a group? 6. According to what axioms can the elements of a group be combined? 7. The order in which the elements are written is not important, is it? 8. What is the main property of an identity element? 9. How many elements may a group have? 10. What is a subgroup? 11. What is the minor subgroup of a group?
Ex. 9. Match the English words and word combinations with the Russian equivalents.
| 1. a curious awareness | a. принадлежать группе |
| 2. a group concert | b. конечная группа |
| 3. to sense implicitly | c. проверить закон |
| 4. a number of entities | d. ряд (некоторое количество) объектов |
| 5. a unique element | e. современная математика |
| 6. to belong to a group | f. понятие группы |
| 7. an identity element | g. удивительная осведомленность |
| 8. a finite group | h. это не имеет значения |
| 9. to verify a law | i. неявно осознавать |
| 10. an infinite number of elements | j. неограниченное количество элементов |
| 11. the contemporary maths | k. однозначный элемент |
| 12. it makes no difference | l. единичный элемент |
Ex. 10. Find out whether the statements are True or False. Use the introductory phrases: