Introduction to Group Theory (cont.)

1.  Product Notation

The product of n elements a1, a2, … an of a set G with multiplication is inductively defined as follows:

This product is also written as a1·a2· … ·an. We have the following generalization of the associative law.

Proposition 1: Let n0, n1, … nr be integers such that 0 = n0 < n1 < … < nr = n. Then

.

This is clear for r = 1. Hence we can assume that it is true for r - 1 and prove it for r factors. The details are left as a homework exercise.

2. Groups

Definition: A set G with a multiplication is called a group if the following (group) axioms are satisfied:

[G1] The set G is not empty.

[G2] If a, b, c Î G, then (ab)c = a (bc).

[G3] There exists in G an element e such that

(1)  For any element a in G, ea = a.

(2)  For any element a in G there exists an element a’ in G such that a’a = e.

In view of axiom [G2] and the general associativity law, we can and will write the product of any finite member of elements of G without inserting parentheses.

Proposition: If G is a group and e the element specified in [G3(1)], then e is an identity element.

Proposition: If G is a group, a an element of G, then a’ (specified in [G3(2)] is its inverse element.

Proposition: If G is a group, then each of the equation ax = b and xa = b in an unknown x has a unique solution.

3. Examples for Groups

The following are groups:

{0,1} with multiplication given by the following table:

· / 0 / 1
0 / 0 / 1
1 / 1 / 0

{0,1,2} with multiplication given by the following table:

· / 0 / 1 / 2
0 / 0 / 1 / 2
1 / 1 / 2 / 0
2 / 2 / 0 / 1

{0,1,2,3} with multiplication given by the following table

· / 0 / 1 / 2 / 3
0 / 0 / 1 / 2 / 3
1 / 1 / 0 / 3 / 2
2 / 2 / 3 / 0 / 1
3 / 3 / 2 / 1 / 0

{0,1,2,3} with multiplication given by the following table

· / 0 / 1 / 2 / 3
0 / 0 / 1 / 2 / 3
1 / 1 / 2 / 3 / 0
2 / 2 / 3 / 0 / 1
3 / 3 / 0 / 1 / 2

4. Subgroups

Definition: Let (G, ·) be a group and (H,°) another one. Then (H,°) is a subgroup of (G, ·) if and only if H Ì G and "a,b Î H: a°b = a·b.

With other words, a subgroup of G is a subset with the same multiplication that is itself a group.

Proposition: Let H Ì G be a subgroup. Then

The identity element of H is the identity element of G .

Proposition: A non-empty subset H of G is a subgroup (with the same multiplication) iff