Mathematical Methods for Physics and Engineering : A Comprehensive Guide

GROUP THEORY

mathematical details, a rotation about axisican be represented by the operator

Ri(θ), and the two rotations are connected by a relationship of the form

Rj(θ)=φ−ij^1 Ri(θ)φij,

in whichφijis the member of the full continuous rotation group that takes axis

iinto axisj.

28.8 Exercises

28.1 For each of the following sets, determine whether they form a group under the op-
eration indicated (where it is relevant you may assume that matrix multiplication
is associative):

(a) the integers (mod 10) under addition;

(b) the integers (mod 10) under multiplication;

(c) the integers 1, 2, 3, 4, 5, 6 under multiplication (mod 7);

(d) the integers 1, 2, 3, 4, 5 under multiplication (mod 6);

(e) all matrices of the form

(

aa−b

0 b

)

,

whereaandbare integers (mod 5) anda=0=b, under matrix multiplica-

tion;

(f) those elements of the set in (e) that are of order 1 or 2 (taken together);

(g) all matrices of the form





100

a 10

bc 1



,

wherea,b,care integers, under matrix multiplication.

28.2 Which of the following relationships betweenXandYare equivalence relations?
Give a proof of your conclusions in each case:

(a) XandYare integers andX−Yis odd;

(b)XandYare integers andX−Yis even;

(c) XandYare people and have the same postcode;

(d)XandYare people and have a parent in common;

(e) XandYare people and have the same mother;

(f) XandYaren×nmatrices satisfyingY=PXQ,wherePandQare elements

of a groupGofn×nmatrices.

28.3 Define a binary operation•on the set of real numbers by

x•y=x+y+rxy,

whereris a non-zero real number. Show that the operation•is associative.

Prove thatx•y=−r−^1 if, and only if,x=−r−^1 ory=−r−^1. Hence prove

that the set of all real numbers excluding−r−^1 forms a group under the oper-

ation•.