Groups and their realizations

Abstract group theory defines relationships among a set of abstract elements in terms of binary operations among the elements of the group. The operations are known as group multiplication.

Formally, a group consists of a set of elements with the following properties:

1. (i) The product of any two elements in the set is a member of the set. Thus the set is closed under all group multiplication operations.

2. (ii) If A, B, and C are elements of the group, then A(BC) = (AB)C. The associative law of multiplication holds; the commutative law of multiplication need not hold.

3. (iii) There is a unit element, an identity element, E such that EA = AE = A.

4. (iv) Each element A has a unique inverse A−1 such that AA−1 = A−1A = E.

A typical abstract group multiplication table is given in Table 2.1, for the group we denote by G6, which consists of six elements.

The convention for such tables is that the ij th element in the table is the product of the ith element labeling the rows and the j th element labeling the columns. From the table we see that AB = D, which means that the operation B followed by the operation A is equivalent to the single operation D. Note that AB = D, but BA = F; thus AB ≠ BA in this case.

Abelian groups If XY = YX for all elements of the group, the group is called Abelian. It is clear from the asymmetry about the diagonal of Table 2.1 that the group G6 is not Abelian.