Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.7 SUBDIVIDING A GROUP

this implies thatZbelongs toSY. These two results together mean that the two

subsetsSXandSYhave the same members and hence are equal.

Now suppose thatSXequalsSY.SinceYbelongs toSYit also belongs toSX

and henceX∼Y. This completes the proof of (i), once the distinct subsets of

typeSXare identified as the classesCi. Statement (ii) is an immediate corollary,

the class in question being identified asSW.

The most important property of an equivalence relation is as follows.

Two different subsetsSXandSYcan have no element in common, and the collection

of all the classesCiis a ‘partition’ ofS, i.e. every element inSbelongs to one, and

only one, of the classes.

To prove this, supposeSXandSYhave an elementZin common; thenX∼Z

andY∼Zand so by the symmetry and transitivity lawsX∼Y.Bytheabove

theorem this impliesSXequalsSY. But this contradicts the fact thatSXandSY

are different subsets. HenceSXandSYcan have no element in common.

Finally, if the elements ofSare used in turn to define subsets and hence classes

inS, every elementUis in the subsetSUthat is either a class already found or

constitutes a new one. It follows that the classes exhaustS,i.e.everyelementis

in some class.

Having established the general properties of equivalence relations, we now turn

to two specific examples of such relationships, in which the general setShas the

more specialised properties of a groupGand the equivalence relation∼is chosen

in such a way that the relatively transparent general results for equivalence

relations can be used to derive powerful, but less obvious, results about the

properties of groups.

28.7.2 Congruence and cosets

As the first application of equivalence relations we now prove Lagrange’s theorem

which is stated as follows.

Lagrange’s theorem.IfGis a finite group of ordergandHis a subgroup ofGof

orderhthengis a multiple ofh.

We take as the definition of∼that, givenXandYbelonging toG,X∼Yif

X−^1 Ybelongs toH. This is the same as saying thatY=XHifor some element

Hibelonging toH; technicallyXandYare said to be left-congruent with respect

toH.

This defines an equivalence relation, since it has the following properties.

(i) Reflexivity:X∼X,sinceX−^1 X=IandIbelongs to any subgroup.

(ii) Symmetry:X∼Yimplies thatX−^1 Ybelongs toHand so, therefore, does

its inverse, sinceHis a group. But (X−^1 Y)−^1 =Y−^1 Xand, as this belongs

toH, it follows thatY∼X.