Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.7 SUBDIVIDING A GROUP

(i) the set of elementsH′inG′that are images of the elements ofGforms a

subgroup ofG′;

(ii) the set of elementsKinGthat are mapped onto the identityI′inG′forms

a subgroup ofG.

As indicated in the previous section, the subgroupKis called thekernelof the

homomorphism.

To prove (i), supposeZandWbelong toH′, withZ=X′andW=Y′,where

XandYbelong toG.Then

ZW=X′Y′=(XY)′

and therefore belongs toH′,and

Z−^1 =(X′)−^1 =(X−^1 )′

and therefore belongs toH′. These two results, together with the fact thatI′

belongs toH′, are enough to establish result (i).

To prove (ii), supposeXandYbelong toK;then

(XY)′=X′Y′=I′I′=I′ (closure),

I′=(XX−^1 )′=X′(X−^1 )′=I′(X−^1 )′=(X−^1 )′

and thereforeX−^1 belongs toK. These two results, together with the fact thatI

belongs toK, are enough to establish (ii). An illustration of this result is provided

by the mapping Φ of →U(1) considered in the previous section. Its kernel

consists of the set of real numbers of the form 2πn,wherenis an integer; it forms

a subgroup ofR, the additive group of real numbers.

In fact the kernelKof a homomorphism is anormalsubgroup ofG.The

defining property of such a subgroup is that for every elementXinGand every

elementYin the subgroup,XY X−^1 belongs to the subgroup. This property is

easily verified for the kernelK,since

(XY X−^1 )′=X′Y′(X−^1 )′=X′I′(X−^1 )′=X′(X−^1 )′=I′.

Anticipating the discussion of subsection 28.7.2, the cosets of a normal subgroup

themselves form a group (see exercise 28.16).

28.7 Subdividing a group

We have already noted, when looking at the (arbitrary) order of headings in a

group table, that some choices appear to make the table more orderly than do

others. In the following subsections we will identify ways in which the elements

of a group can be divided up into sets with the property that the members of any

one set are more like the other members of the set, in some particular regard,