The group concept

The concept of a group is abstracted from arithmetic. A group consists of:

1. A set G.

2. an operation (*) which assigns to any elements x and y of G an element x*y which also belongs to G.

This operation is required to satisfy three laws:

3. The operation is associative: for any x, y, z  G we have

x* (y*z) = (x*y)*z

4. There is an identity element I  G such that

I*x = x = x*I

for any x  G.

5. There are inverses: for any x  G there exists x' G such that

x*x' = I = x'*x.

Groups can arise in many quite distinct situations. Here are some examples:

1. Let G be the set of integers: G=Z. Then (1) holds. Let * be the operation + of addition. Then (2) holds because if a and b are integers then a + b is an integer. Condition (3) is law (1) of arithmetics, (4) is law (3) (with 0 playing the part of I), (5) is law (4).

It should be emphasized that the failure of any of the five conditions means that we do not have a group.

If we took G to be the set of integers between -10 and 10, and * to be addition, then (2) is violated: 6+6 is not an element of G.

The set of integers greater than 1, under the operation of addition, has no element satisfying (4).

The set of integers, under the operation of subtraction, violates condition (3) because subtraction is not associative:

(2-3) – 5 = - 6 ≠ 4 = 2 – (3 – 5).

The set of all rationals, under multiplication, is not a group. The only element that we can find for I is 1, and then we cannot find an element 0' such that 0'0 = 1; because for any rational r we have r x 0 = 0, so 0'0 = 0 , not 1.

So none of these define groups.

Let us say a few words about the operation *. Given any pair of elements (x, y) where x, y _ G we obtain a unique element x * y of G. This means that * defines a function whose domain is the set G x G of pairs (x, y), and whose range is G. An operation may be defined as a function

* : G x G → G,

having agreed that x * y is shorthand for * (x, y). Once we do things this way, condition (II) is automatic, and may be omitted except that we have to check, in any particular case, that * really is a function from G x G into G.

Having understood the ideas, we can simplify our notation. Instead of x * y we can write x y remembering that this need not be ordinary multiplication and it then becomes natural to write x' = x^-1. If you use this notation working in the group of integers under addition, then x y means x + y and x^-1 means –x. It is important not to get confused!

Grammar and vocabulary exercises

I. Read and translate the following sentences paying attention to “once” which is used as a conjunction.

1. Once we know what the answer is, we must verify it. 2. The proofs are not hard to understand once you see them. 3. Once some power becomes equal to1, the sequence must repeat. 4. Once we have chosen one set, we can use it instead of V. 5. Once I have got over this, there remains the problem of proving that the laws are true. 6. Once we do things this way, our condition is automatic. 7. Once we understand the idea, we can simplify our notation.

II. Read and translate the following sentences in which Participle I is used as an attribute.

1. The objects belonging to the set are the elements or members of the set. 2. S is a set consisting of a pencil and an eraser. 3. A line passing through a point inside a triangle must cut the triangle somewhere. 4. A line is the set of all pairs (x, y) satisfying its equation. 5. A set having operations of addition and multiplication which satisfy the nine laws of algebra is called a field. 6. The group of integers under addition has subgroups comprising all even integers. 7. The puzzle cannot be solved using lines that do not cross. 8. We get a new sequence containing every algebraic number. 9. An equation is a statement showing the equality of two quantities. 10. Fractions having different forms but equal values are equivalent fractions.

III. Read and translate the following sentences in which Participle I is used as an adverbial modifier.

1. Certain properties of the real world can be described using numbers. 2. Many of the concepts can be vividly illustrated using simple apparatus. 3. When using the curly bracket notation the elements are thought of as occurring once only in the set. 4. Proceeding in this way, we can set up the whole of Euclidean geometry as a part of set theory. 5. Having established that there is just one empty set, we can give it a symbol. 6. Having started from a system of axioms, we then could make certain logical deductions. 7There are many useful functions which are not easily defined using formulas. 8. Using algebra we can reduce complex problems to simple formulas. 9. When finding the product of multinomials we make use of the distributive law. 10. When speaking of quantities we shall have in view their numerical values.