Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

(iii) Transitivity:X∼YandY∼Zimply thatX−^1 YandY−^1 Zbelong toH

and so, therefore, does their product (X−^1 Y)(Y−^1 Z)=X−^1 Z,fromwhich

it follows thatX∼Z.

With∼proved as an equivalence relation, we can immediately deduce that it

dividesGinto disjoint (non-overlapping) classes. For this particular equivalence

relation the classes are called theleft cosetsofH. Thus each element ofGis in

one and only one left coset ofH. The left coset containing any particularXis

usually writtenXH, and denotes the set of elements of the formXHi(one of

which isXitself sinceHcontains the identity element); it must containhdifferent

elements, since if it did not, and two elements were equal,

XHi=XHj,

we could deduce thatHi=Hjand thatHcontained fewer thanhelements.

From our general results about equivalence relations it now follows that the

left cosets ofHare a ‘partition’ ofGinto a number of sets each containingh

members. Since there aregmembers ofGand each must be in just one of the

sets, it follows thatgis a multiple ofh. This concludes the proof of Lagrange’s

theorem.

The number of left cosets ofHinGis known as theindexofHinGand is

written [G:H]; numerically the index =g/h. For the record we note that, for

the trivial subgroupI, which contains only the identity element, [G:I]=gand

that, for a subgroupJof subgroupH,[G:H][H:J]=[G:J].

The validity ofLagrange’s theoremwas established above using the far-reaching

properties of equivalence relations. However, for this specific purpose there is a

more direct and self-contained proof, which we now give.

LetXbe some particular element of a finite groupGof orderg,andHbe a

subgroup ofGof orderh, with typical elementYi. Consider the set of elements

XH≡{XY 1 ,XY 2 ,...,XYh}.

This set containshdistinct elements, since if any two were equal, i.e.XYi=XYj

withi=j, this would contradict the cancellation law. As we have already seen,

the set is called a left coset ofH.

We now prove three simple results.

• Two cosets are either disjoint or identical.Suppose cosetsX 1 HandX 2 Hhave

an element in common, i.e.X 1 Y 1 =X 2 Y 2 for someY 1 ,Y 2 inH.ThenX 1 =

X 2 Y 2 Y 1 −^1 , and sinceY 1 andY 2 both belong toHso doesY 2 Y 1 −^1 ; thusX 1

belongs to the left cosetX 2 H. SimilarlyX 2 belongs to the left cosetX 1 H.

Consequently, either the two cosets are identical or it was wrong to assume

that they have an element in common.