Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

The multiplication table for this set of six functions has all the necessary proper-

ties to show that they form a group. Further, if the symbolsf 1 ,f 2 ,f 3 ,f 4 ,f 5 ,f 6 are

replaced byI, A, B, C, D, Erespectively the table becomes identical to table 28.8.

This justifies our earlier claim that this group of functions, with argument sub-

stitution as the law of combination, is isomorphic to the group of reflections and

rotations of an equilateral triangle.

28.4 Permutation groups

The operation of rearrangingndistinct objects amongst themselves is called a

permutationof degreen, and since many symmetry operations on physical systems

can be viewed in that light, the properties of permutations are of interest. For

example, the symmetry operations on an equilateral triangle, to which we have

already given much attention, can be considered as the six possible rearrangements

of the marked corners of the triangle amongst three fixed points in space, much

as in the diagrams used to compute table 28.7. In the same way, the symmetry

operations on a cube can be viewed as a rearrangement of its corners amongst

eight points in space, albeit with many constraints, or, with fewer complications,

as a rearrangement of its body diagonals in space. The details will be left until

we review the possible finite groups more systematically.

The notations and conventions used in the literature to describe permutations

are very varied and can easily lead to confusion. We will try to avoid this by using

lettersa,b,c,...(rather than numbers) for the objects that are rearranged by a

permutation and by adopting, before long, a ‘cycle notation’ for the permutations

themselves. It is worth emphasising that it is thepermutations,i.e.theactsof

rearranging, and not the objects themselves (represented by letters) that form

the elements of permutation groups. The complete group of all permutations of

degreenis usually denoted bySnor Σn. The number of possible permutations of

degreenisn!, and so this is the order ofSn.

Suppose the ordered set of six distinct objects{abcdef}is rearranged by

some process into{befadc}; then we can represent this mathematically as

θ{abcdef}={befadc},

whereθis a permutation of degree 6. The permutationθcan be denoted by

[256143],sincethefirstobject,a, is replaced by the second,b, the second

object,b, is replaced by the fifth,e, the third by the sixth,f,etc.Theequation

can then be written more explicitly as

θ{abcdef}=[256143]{abcdef}={befadc}.

Ifφis a second permutation, also of degree 6, then the obvious interpretation of

the productφ•θof the two permutations is

φ•θ{abcdef}=φ(θ{abcdef}).