Mathematical Methods for Physics and Engineering - Matematica.NET

28.3 NON-ABELIAN GROUPSI A B C D EI I A B C D EA A B I E C DB B I A D E CC C D E I A BD D E C B I AE E C D A B ITable 28.8 The group table, under matrix multiplication, for the set M ofsix orthogonal 2 × 2 matrices given by (28.13).The similarity to table 28.7 is striking. If {R,R ′ ,K,L,M} of that table arereplaced by {A, B, C, D, E} respectively, the two tables are identical, without eventhe need to reshuffle the rows and columns. The two groups, one of reflectionsand rotations of an equilateral triangle, the other of matrices, are isomorphic.Our second example of a group isomorphic to the same rotation–reflectiongroup is provided by a set of functions of an undetermined variable x. Thefunctions are as follows:f 1 (x) =x, f 2 (x) =1/(1 − x), f 3 (x) =(x − 1)/x,f 4 (x) =1/x, f 5 (x) =1− x, f 6 (x) =x/(x − 1),and the law of combination isf i (x) • f j (x) =f i (f j (x)),i.e. the function on the right acts as the argument of the function on the left toproduce a new function of x. It should be emphasised that it is the functionsthat are the elements of the group. The variable x is the ‘system’ on which theyact, and plays much the same role as the triangle does in our first example of anon-Abelian group.To show an explicit example, we calculate the product f 6 • f 3 . The productwill be the function of x obtained by evaluating y/(y − 1), when y is set equal to(x − 1)/x. Explicitly(x − 1)/xf 6 (f 3 )=(x − 1)/x − 1 =1− x = f 5(x).Thus f 6 • f 3 = f 5 . Further examples aref 2 • f 2 =11 − 1/(1 − x) = x − 1 = f 3 ,xandf 6 • f 6 =x/(x − 1)x/(x − 1) − 1 = x = f 1. (28.14)1055