Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

1357

1 1357

3 317 5

5 5713

7 7531

Table 28.1 The table of products for the elements of the groupS={ 1 , 3 , 5 , 7 }

under multiplication (mod 8).

usual to list the elements in the same order in both the vertical and horizontal

headings in any one table. The actual order of the elements in the common list,

whilst arbitrary, is normally chosen to make the table have as much symmetry as

possible. This is initially a matter of convenience, but, as we shall see later, some

of the more subtle properties of groups are revealed by putting next to each other

elements of the group that are alike in certain ways.

Some simple general properties of group multiplication tables can be deduced

immediately from the fact that each row or column constitutes the elements of

the group.

(i) Each element appears once and only once in each row or column of the

table; this must be so sinceG•X=G(the permutation law) holds.

(ii) The inverse of any elementY can be found by looking along the row

in whichYappears in the left-hand column (theYth row), and noting

the elementZat the head of the column (theZth column) in which

the identity appears as the table entry. An immediate corollary is that

whenever the identity appears on the leading diagonal, it indicates that

the corresponding header element is of order 2 (unless it happens to be

the identity itself).

(iii) For any Abelian group the multiplication table is symmetric about the

leading diagonal.

To get used to the ideas involved in using group multiplication tables, we now

consider two more sets of integers under multiplication (modN):

S′={ 1 , 5 , 7 , 11 } under multiplication (mod 24), and

S′′={ 1 , 2 , 3 , 4 } under multiplication (mod 5).

These have group multiplication tables 28.2(a) and (b) respectively, as the reader

should verify.

If tables 28.1 and 28.2(a) for the groupsSandS′are compared, it will be seen

that they have essentially the same structure, i.e if the elements are written as

{I, A, B, C}in both cases, then the two tables are each equivalent to table 28.3.

ForS,I=1,A=3,B=5,C= 7 and the law of combination is multiplication

(mod 8), whilst forS′,I=1,A=5,B=7,C= 11 and the law of combination