Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.3 NON-ABELIAN GROUPS

As a first example we consider again as elements of a group the two-dimensional

operations which transform an equilateral triangle into itself (see the end of

subsection 28.1.1). It has already been shown that there are six such operations:

the null operation, two rotations (by 2π/3and4π/3 about an axis perpendicular

to the plane of the triangle) and three reflections in the perpendicular bisectors

of the three sides. To abbreviate we will denote these operations by symbols as

follows.

(i)Iis the null operation.

(ii)RandR′are (clockwise) rotations by 2π/3and4π/3 respectively.

(iii)K,L,Mare reflections in the three lines indicated in figure 28.2.

Some products of the operations of the formX•Y(where it will be recalled

that the symbol•means that the second operationXis carried out on the system

resulting from the application of the first operationY) are easily calculated:

R•R=R′,R′•R′=R, R•R′=I=R′•R

(28.12)

K•K=L•L=M•M=I.

Others, such asK•M, are more difficult, but can be found by a little thought,

or by making a model triangle or drawing a sequence of diagrams such as those

following.

x

x x

x

K•M =K = = R′

showing thatK•M=R′. In the same way,

x x x x

M•K = M = = R

shows thatM•K=R,and

x x xx

R•L = R = = K

shows thatR•L=K.

Proceeding in this way we can build up the complete multiplication table

(table 28.7). In fact, it is not necessary to draw any more diagrams, as all

remaining products can be deduced algebraically from the three found above and