Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

than they are like any element that does not belong to the set. We will find that

these divisions will be such that the group ispartitioned, i.e. the elements will be

divided into sets in such a way that each element of the group belongs to one,

and only one, such set.

We note in passing that the subgroups of a group donotform such a partition,

not least because the identity element is in every subgroup, rather than being in

precisely one. In other words, despite the nomenclature, a group is not simply the

aggregate of its proper subgroups.

28.7.1 Equivalence relations and classes

We now specify in a more mathematical manner what it means for two elements

of a group to be ‘more like’ one another than like a third element, as mentioned

in section 28.2. Our introduction will apply to any set, whether a group or not,

but our main interest will ultimately be in two particular applications to groups.

We start with the formal definition of an equivalence relation.

Anequivalence relationon a setSis a relationshipX∼Y, between two

elementsXandYbelonging toS, in which the definition of the symbol∼must

satisfy the requirements of

(i) reflexivity,X∼X;

(ii) symmetry,X∼YimpliesY∼X;

(iii) transitivity,X∼YandY∼ZimplyX∼Z.

Any particular two elements either satisfy or do not satisfy the relationship.

The general notion of an equivalence relation is very straightforward, and

the requirements on the symbol∼seem undemanding; but not all relationships

qualify. As an example within the topic of groups, if it meant ‘has the same

order as’ then clearly all the requirements would be satisfied. However, if it meant

‘commutes with’ then it would not be an equivalence relation, since althoughA

commutes withI,andIcommutes withC, this does not necessarily imply thatA

commutes withC, as is obvious from table 28.8.

It may be shown that an equivalence relation onSdivides upSintoclassesCi

such that:

(i)XandYbelong to the same class if, and only if,X∼Y;

(ii) every elementWofSbelongs to exactly one class.

This may be shown as follows. LetXbelong toS, and define the subsetSXof

Sto be the set of all elementsUofSsuch thatX∼U. Clearly by reflexivity

Xbelongs toSX. Suppose first thatX∼Y, and letZbe any element ofSY.

ThenY∼Z, and hence by transitivityX∼Z, which means thatZbelongs to

SX. Conversely, since the symmetry law givesY∼X,ifZbelongs toSXthen