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 Î G we have

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

(4) There is an identity element I Î G such that

I*x = x = x*I

for any x Î G.

(5) There are inverses: for any x Î G there exists x'Î 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 _Î 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!

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