An abstract group G is a set of elements for which a law of composition or "product" is defined.

For illustrative purposes let us consider a simple set of three elements {E, A, B}. A law of composition for these three elements can be expressed in the form of a multiplication table, see Table 1. In position i, j of Table 1. we find the product of element i and element j with the numbering 1 ^ E,2 ^ A,3 ^ B. From the table we can read out that B = AA because element 2, 2 is B and the third line contains the products AE, AA, AB. The table is symmetric therefore AB = BA. Such a group is formed for example by the even permutations of three objects: E = (a, b, c), A = (c, a, b), B = (b, c, a). The multiplication table reflects four necessary

Table 1. Multiplication table for elements { E, A, B}

conditions that a set of elements must satisfy to form a group G. These four conditions are:

1. The product of any two elements of G is also an element of G. Such as for example AB = E.

2. Multiplication is associative. For example (AB)E = A(BE).

3. G contains a unique element E called the identity element, such that for example AE = EA = A and the same holds for every element of G.

4. For every element in G there exists another element in G, such that their product is the identity element. In our example AB = E therefore B is called the inverse of A and is denoted B = A-1.

The application of group theory to physical problems arises from the fact that many characteristics of physical problems, in particular symmetries and invariance, conform to the definition of groups, and thereby allows us to bring to bear on the solution of physical problems the machinery of abstract group theory.

For example, if we consider a characteristic of an equilateral triangle we observe the following with regard to the counter clockwise rotations by 120 degrees. Let us give the operations the following symbols: E-no rotations, Сэ-rotation by 120°, C3C3 = C2-rotation by 240°. The group operation is the sequential application of these operations, the leftmost operator should be applied first. The reader can easily check the multiplication table 2. applies to the

Table 2. Multiplication table of G = {E, C3, C^}

Table 3. Multiplication table of G3

Table 4. Multiplication table of a permutation group

group G = {E, C3,C2}. We see immediately that the multiplication table of the rotations of an equilateral triangle is identical to the multiplication table of the previous abstract group G = {E, C3, C2}. Thus there is a one-to-one correspondence (called isomorphism) between the abstract group of the previous example, and its rules.