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 to various 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 to form 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 group if 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^-1 such 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, -i form a finite group under multiplication. Neither does it need to use ordinary multiplication as the form combination. 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.