Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

each number appears once and only once in the representation of any particular

permutation.

Theorder of any permutationof degreenwithin the groupSncanbereadoff

from the cyclic representation and is given by the lowest common multiple (LCM)

of the lengths of the cycles. ThusIhas order 1, as it must, and the permutation

θdiscussed above has order 4 (the LCM of 4 and 2).

Expressed in cycle notation our second permutationφis (3)(1 4 6)(2 5), and

the productφ•θis calculated as

(3)(1 4 6)(2 5)•(1 2 5 4)(3 6){abcdef}= (3)(1 4 6)(2 5){befadc}

={adfceb}

= (1)(5)(2 4 3 6){abcdef}.

i.e. expressed as a relationship amongst the elements of the group of permutations

of degree 6 (not yet proved as a group, but reasonably anticipated), this result

reads

(3)(1 4 6)(2 5)•(1 2 5 4)(3 6) = (1)(5)(2 4 3 6).

We note, for practice, thatφhas order 6 (the LCM of 1, 3, and 2) and that the

productφ•θhas order 4.

The number of elements in the groupSnof all permutations of degreenis

n! and clearly increases very rapidly asnincreases. Fortunately, to illustrate the

essential features of permutation groups it is sufficient to consider the casen=3,

which involves only six elements. They are as follows (with labelling which the

reader will by now recognise as anticipatory):

I= (1)(2)(3) A=(123) B=(132)

C= (1)(2 3) D= (3)(1 2) E= (2)(1 3)

It will be noted thatAandBhave order 3, whilstC,DandEhave order 2. As

perhaps anticipated, their combination products are exactly those corresponding

to table 28.8,I,C,DandEbeing their own inverses. For example, putting in all

steps explicitly,

D•C{abc}= (3)(1 2)•(1)(2 3){abc}

= (3)(12){acb}

={cab}

=(321){abc}

=(132){abc}

=B{abc}.

In brief, the six permutations belonging toS 3 form yet another non-Abelian group

isomorphic to the rotation–reflection symmetry group of an equilateral triangle.