Mathematical Methods for Physics and Engineering - Matematica.NET

28.2 FINITE GROUPS28.2 Finite groupsWhilst many properties of physical systems (e.g. angular momentum) are relatedto the properties of infinite, and, in particular, continuous groups, the symmetryproperties of crystals and molecules are more intimately connected with those offinite groups. We therefore concentrate in this section on finite sets of objects thatcan be combined in a way satisfying the group postulates.Although it is clear that the set of all integers does not form a group underordinary multiplication, restricted sets can do so if the operation involved is multiplication(mod N) for suitable values of N; this operation will be explained below.As a simple example of a group with only four members, consider the set Sdefined as follows:S = {1, 3, 5, 7} under multiplication (mod 8).To find the product (mod 8) of any two elements, we multiply them together inthe ordinary way, and then divide the answer by 8, treating the remainder afterdoing so as the product of the two elements. For example, 5 × 7 = 35, which ondividing by 8 gives a remainder of 3. Clearly, since Y × Z = Z × Y , the full setof different products is1 × 1=1, 1 × 3=3, 1 × 5=5, 1 × 7=7,3 × 3=1, 3 × 5=7, 3 × 7=5,5 × 5=1, 5 × 7=3,7 × 7=1.The first thing to notice is that each multiplication produces a member of theoriginal set, i.e. the set is closed. Obviously the element 1 takes the role of theidentity, i.e. 1 × Y = Y for all members Y of the set. Further, for each element Yof the set there is an element Z (equal to Y , as it happens, in this case) such thatY × Z = 1, i.e. each element has an inverse. These observations, together with theassociativity of multiplication (mod 8), show that the set S is an Abelian groupof order 4.It is convenient to present the results of combining any two elements of agroup in the form of multiplication tables – akin to those which used to appear inelementary arithmetic books before electronic calculators were invented! Writtenin this much more compact form the above example is expressed by table 28.1.Although the order of the two elements being combined does not matter herebecause the group is Abelian, we adopt the convention that if the product in ageneral multiplication table is written X • Y then X is taken from the left-handcolumn and Y is taken from the top row. Thus the bold ‘7’ in the table is theresult of 3 × 5, rather than of 5 × 3.Whilst it would make no difference to the basic information content in a tableto present the rows and columns with their headings in random orders, it is1049