Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.1 GROUPS

L M

K

Figure 28.2 Reflections in the three perpendicular bisectors of the sides of

an equilateral triangle take the triangle into itself.

(iii) Forming the product of each element ofGwith a fixed elementXofG

simply permutes the elements ofG; this is often written symbolically as

G•X=G. If this were not so, andX•YandX•Zwere not different

even thoughYandZwere, application of the cancellation law would lead

to a contradiction. This result is called thepermutation law.

In any finite group of orderg, any elementXwhen combined with itself to

form successivelyX^2 =X•X,X^3 =X•X^2 ,...will, after at mostg−1such

combinations, produce the group identityI.OfcourseX^2 ,X^3 ,...are some of

the original elements of the group, and not new ones. If the actual number of

combinations needed ism−1, i.e.Xm=I,thenmis called theorder of the element

XinG. The order of the identity of a group is always 1, and that of any other

element of a group that is its own inverse is always 2.

Determine the order of the group of (two-dimensional) rotations and reflections that take

a plane equilateral triangle into itself and the order of each of the elements. The group is

usually known as 3 m(to physicists and crystallographers) orC 3 v(to chemists).

There are two (clockwise) rotations, by 2π/3and4π/3, about an axis perpendicular to
the plane of the triangle. In addition, reflections in the perpendicular bisectors of the three
sides (see figure 28.2) have the defining property. To these must be added the identity
operation. Thus in total there are six distinct operations and sog= 6 for this group.
To reproduce the identity operation either of the rotations has to be applied three times,
whilst any of the reflections has to be applied just twice in order to recover the original
situation. Thus each rotation element of the group has order 3, and each reflection element
has order 2.

A so-calledcyclic groupis one for which all members of the group can be

generated from just one elementX(say). Thus a cyclic group of ordergcan be

written as

G=

{

I, X, X^2 ,X^3 ,...,Xg−^1

}

.