FOR SCIENCE STUDENTS

Solutions Manual to accompany AN

INTRODUCTION

TO GROUPS

AND THEIR

MATRICES

FOR SCIENCE STUDENTS

KOLENKOW 1 / 4

1.1 ApplyTx=x.Tx=x Te x =e (x) =e x Here is another approach. Apply the operator toe x expressed in series form.Te x =T

1+x+ x 2 2!+ x 3 3!

+: : :

!=

1x+ (x) 2 2!+ (x) 3 3!

+: : :

!=

1x+ x 2 2!

x 3 3!

+: : :

!=e x The operator has changed the sign of the odd terms. 2 / 4

2FUNDAMENTAL CONCEPTS

1.2 ApplyTx=x.Tcos (x)=cos (x) =cos (x) This result can also be understood by considering the series expansion of cos (x).cos (x)=1 x 2 2!+ x 4 4!

x 6 6!

+: : :

Tcos (x)=cos (x) =1 (x) 2 2!+ (x) 4 4!

(x) 6 6!

+: : :

=1 x 2 2!+ x 4 4!

x 6 6!

+: : :

=cos (x) The series expansion of cos (x) contains only even terms, soTx=xchanges nothing.

1.3 According to the group axioms

(AB)(AB)

1 =E Multiply byA 1 from the left.A 1

AB(AB)

1 =A 1 E B(AB) 1 =A 1 Multiply byB 1 from the left.B 1 B(AB) 1 =B 1 A 1 (AB) 1 =B 1 A

1 3 / 4

FUNDAMENTAL CONCEPTS3

1.4 Check whether the axioms that dene a group (Sec. (1.3) of the text) are satised for positive and negative real integers under addition.The sum of two integers is also an integer, a member of the set, satisfying axiom (ii) that the group operation does not result in an element outside the set.The sum of an integerNand 0 is equal toN, so 0 acts as the identity element under addition, satisfying axiom (iii) that the group must have an identity element.The sum of an integerNand its negativeNisNN, equal to the identity element

• Every integerNhas an inverse elementNunder addition, so axiom (iv) is sat-

ised that the group must contain an inverse under the group operation for every element.

Trivially, the addition of integers is associative:N1+(N2+N3)=(N1+N2)+N3,

satisfying axiom(v).

1.5 Check whether the axioms that dene a group (Sec. (1.3) of the text) are satised by the set of real integers under multiplication.The product of two integers is also an integer, a member of the set, satisfying axiom (ii) that the group operation does not result in an element outside the set.The product of an integerNand 1 is equal toN, so 1 acts as the identity element under multiplication, satisfying axiom (iii) that the group must have an identity el- ement.Axiom (iv) requires thatNN 1 =1. It follows thatN 1 = 1 N . The inverse of a real integer under multiplication is not an integer except for 1, so the real integers are not a group because the set of real integers does not include the inverses under multiplication.

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