Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.7 SUBDIVIDING A GROUP

(iii) In any groupGthe setSof elements in classes by themselves is an Abelian

subgroup (known as thecentreofG). We have shown thatIbelongs toS,

and so if, further,XGi=GiXandYGi=GiYfor allGibelonging toG

then:

(a) (XY)Gi=XGiY=Gi(XY),i.e. the closure ofS,and

(b)XGi=GiXimpliesX−^1 Gi=GiX−^1 , i.e. the inverse ofXbelongs

toS.

HenceSis a group, and clearly Abelian.

Yet again for illustration purposes, we use the six-element group that has

table 28.8 as its group table.

Find the conjugacy classes of the groupGhaving table 28.8 as its multiplication table.

As always,Iis in a class by itself, and we need consider it no further.
Consider next the results of formingX−^1 AX,asXruns through the elements ofG.

I−^1 AI A−^1 AA B−^1 AB C−^1 AC D−^1 AD E−^1 AE

=IA =IA =AI =CE =DC =ED

=A =A =A =B =B =B

OnlyAandBare generated. It is clear that{A, B}is one of the conjugacy classes ofG.
This can be verified by forming all elementsX−^1 BX; again onlyAandBappear.
We now need to pick an element not in the two classes already found. Suppose we
pickC.JustasforA, we computeX−^1 CX,asXruns through the elements ofG.The
calculations can be done directly using the table and give the following:

X :IABCDE

X−^1 CX :CEDCED

ThusC,DandEbelong to the same class. The group is now exhausted, and so the three
conjugacy classes are

{I}, {A, B}, {C, D, E}.

In the case of this small and simple, but non-Abelian, group, only the identity

is in a class by itself (i.e. onlyIcommutes with all other elements). It is also the

only member of the centre of the group.

Other areas from which examples of conjugacy classes can be taken include

permutations and rotations. Two permutations can only be (but are not nec-

essarily) in the same class if their cycle specifications have the same structure.

For example, inS 5 the permutations (1 3 5)(2)(4) and (2 5 3)(1)(4) could be in

the same class as each other but not in the class that contains (1 5)(2 4)(3). An

example of permutations with the same cycle structure yet in different conjugacy

classes is given in exercise 29. 10.

In the case of the continuous rotation group, rotations by the same angleθ

about any two axes labellediandjare in the same class, because the group

contains a rotation that takes the first axis into the second. Without going into