Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

(a)

IAB

I IAB

A AB I

B BIA

(b)

IABCD

I IABCD

A ABCD I

B BCD I A

C CD I AB

D DI ABC

Table 28.10 The group tables of two cyclic groups, of orders 3 and 5. They

have no proper subgroups.

It will be clear that for a cyclic groupGrepeated combination of any element

with itself generates all other elements ofG, before finally reproducing itself. So,

for example, in table 28.10(b), starting with (say)D, repeated combination with

itself produces, in turn,C,B,A,Iand finallyDagain. As noted earlier, in any

cyclic groupGevery element, apart from the identity, is of orderg, the order of

the group itself.

The two tables shown are for groups of orders 3 and 5. It will be proved in

subsection 28.7.2 that the order of any group is a multiple of the order of any of

its subgroups (Lagrange’s theorem), i.e. in our general notation,gis a multiple

ofh. It thus follows that a group of orderp,wherepis any prime, must be cyclic

and cannot have any proper subgroups. The groups for which tables 28.10(a) and

(b) are the group tables are two such examples. Groups of non-prime order may

(table 28.3) or may not (table 28.6) have proper subgroups.

As we have seen, repeated multiplication of an elementX(not the identity)

by itself will generate a subgroup{X, X^2 ,X^3 ,...}. The subgroup will clearly be

Abelian, and ifXis of orderm,i.e.Xm=I, the subgroup will havemdistinct

members. Ifmis less thang– though, in view of Lagrange’s theorem,mmust

be a factor ofg– the subgroup will be a proper subgroup. We can deduce, in

passing, that the order of any element of a group is an exact divisor of the order

of the group.

Some obvious properties of the subgroups of a groupG, which can be listed

without formal proof, are as follows.

(i) The identity element ofGbelongs to every subgroupH.

(ii) If elementXbelongs to a subgroupH, so doesX−^1.

(iii) The set of elements inGthat belong to every subgroup ofGthemselves

form a subgroup, though this may consist of the identity alone.

Properties of subgroups that need more explicit proof are given in the follow-

ing sections, though some need the development of new concepts before they

can be established. However, we can begin with a theorem, applicable to all

homomorphisms, not just isomorphisms, that requires no new concepts.

Let Φ :G→G′be a homomorphism ofGintoG′;then