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Group Theory With Applications in Chemical Physics
Group theory is widely used in many branches of physics and chemistry, and today it may
be considered as an essential component in the training of both chemists and physicists.
This book provides a thorough, self-contained introduction to the fundamentals of group
theory and its applications in chemistry and molecular and solid state physics. The first half
of the book, with the exception of a few marked sections, focuses on elementary topics. The
second half (Chapters 11–18) deals with more advanced topics which often do not receive
much attention in introductory texts. These include the rotation group, projective representations, space groups, and magnetic crystals. The book includes numerous examples,
exercises, and problems, and it will appeal to advanced undergraduates and graduate
students in the physical sciences. It is well suited to form the basis of a two-semester
course in group theory or for private study.
P R O F E S S O R P . W . M . J A C O B S is Emeritus Professor of Physical Chemistry at the
University of Western Ontario, where he taught widely in the area of physical chemistry,
particularly group theory. He has lectured extensively on his research in North America,
Europe, and the former USSR. He has authored more than 315 publications, mainly in solid
state chemistry and physics, and he was awarded the Solid State Medal of the Royal Society
of Chemistry.
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Group Theory With Applications
in Chemical Physics
P. W. M. JACOBS
The University of Western Ontario
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cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521642507
© P. W. M. Jacobs 2005
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
isbn-13
isbn-10
978-0-511-12913-1 eBook (EBL)
0-511-12913-0 eBook (EBL)
isbn-13
isbn-10
978-0-521-64250-7 hardback
0-521-64250-7 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
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To MFM
and to all those who love group theory
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Contents
Preface
Notation and conventions
page xi
xiii
1 The elementary properties of groups
1.1 Definitions
1.2 Conjugate elements and classes
1.3 Subgroups and cosets
1.4 The factor group
1.5 Minimal content of Sections 1.6, 1.7, and 1.8
1.6 Product groups
1.7 Mappings, homomorphisms, and extensions
1.8 More about subgroups and classes
Problems
1
1
5
6
8
12
15
17
18
22
2 Symmetry operators and point groups
2.1 Definitions
2.2 The multiplication table – an example
2.3 The symmetry point groups
2.4 Identification of molecular point groups
Problems
23
23
32
36
48
50
3 Matrix representatives
3.1 Linear vector spaces
3.2 Matrix representatives of operators
3.3 Mappings
3.4 Group representations
3.5 Transformation of functions
3.6 Some quantum mechanical considerations
Problems
53
53
55
60
62
62
67
68
4 Group representations
4.1 Matrix representations
4.2 Irreducible representations
4.3 The orthogonality theorem
4.4 The characters of a representation
4.5 Character tables
4.6 Axial vectors
70
70
72
73
74
80
82
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viii
Contents
4.7 Cyclic groups
4.8 Induced representations
Problems
86
88
95
5
Bases of representations
5.1 Basis functions
5.2 Construction of basis functions
5.3 Direct product representations
5.4 Matrix elements
Problems
96
96
97
99
101
105
6
Molecular orbitals
6.1 Hybridization
6.2 p Electron systems
6.3 Equivalent bond orbitals
6.4 Transition metal complexes
Problems
106
106
109
114
117
129
7
Crystal-field theory
7.1 Electron spin
7.2 Spherical symmetry
7.3 Intermediate crystal field
7.4 Strong crystal fields
Problems
131
131
132
134
139
146
8
Double groups
8.1 Spin–orbit coupling and double groups
8.2 Weak crystal fields
Problems
148
148
152
154
9
Molecular vibrations
9.1 Classification of normal modes
9.2 Allowed transitions
9.3 Inelastic Raman scattering
9.4 Determination of the normal modes
Problems
156
156
158
161
162
168
10 Transitions between electronic states
10.1 Selection rules
10.2 Vibronic coupling
10.3 Charge transfer
Problems
171
171
173
178
181
11 Continuous groups
11.1 Rotations in R2
11.2 The infinitesimal generator for SO(2)
11.3 Rotations in R3
11.4 The commutation relations
11.5 The irreducible representations of SO(3)
182
182
183
184
187
192
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Contents
ix
11.6 The special unitary group SU(2)
11.7 Euler parameterization of a rotation
11.8 The homomorphism of SU(2) and SO(3)
Problems
200
205
208
216
12 Projective representations
12.1 Complex numbers
12.2 Quaternions
12.3 Geometry of rotations
12.4 The theory of turns
12.5 The algebra of turns
12.6 Projective representations
12.7 Improper groups
12.8 The irreducible representations
Problems
218
218
220
222
225
228
232
240
243
250
13 Time-reversal symmetry
13.1 Time evolution
13.2 Time reversal with neglect of electron spin
13.3 Time reversal with spin–orbit coupling
13.4 Co-representations
Problems
252
252
253
254
257
264
14 Magnetic point groups
14.1 Crystallographic magnetic point groups
14.2 Co-representations of magnetic point groups
14.3 Clebsch–Gordan coefficients
14.4 Crystal-field theory for magnetic crystals
Problems
265
265
267
277
280
281
15 Physical properties of crystals
15.1 Tensors
15.2 Crystal symmetry: the direct method
15.3 Group theory and physical properties of crystals
15.4 Applications
15.5 Properties of crystals with magnetic point groups
Problems
282
282
286
288
293
303
305
16 Space groups
16.1 Translational symmetry
16.2 The space group of a crystal
16.3 Reciprocal lattice and Brillouin zones
16.4 Space-group representations
16.5 The covering group
16.6 The irreducible representations of G
16.7 Herring method for non-symmorphic space groups
16.8 Spinor representations of space groups
Problems
307
307
314
324
331
336
337
344
351
355
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Contents
17 Electronic energy states in crystals
17.1 Translational symmetry
17.2 Time-reversal symmetry
17.3 Translational symmetry in the reciprocal lattice representation
17.4 Point group symmetry
17.5 Energy bands in the free-electron approximation: symmorphic
space groups
17.6 Free-electron states for crystals with non-symmorphic
space groups
17.7 Spinor representations
17.8 Transitions between electronic states
Problems
357
357
357
358
359
378
383
384
390
18 Vibration of atoms in crystals
18.1 Equations of motion
18.2 Space-group symmetry
18.3 Symmetry of the dynamical matrix
18.4 Symmetry coordinates
18.5 Time-reversal symmetry
18.6 An example: silicon
Problems
391
391
394
398
401
404
406
412
365
Appendices
A1 Determinants and matrices
413
A2 Class algebra
434
A3 Character tables for point groups
447
A4 Correlation tables
467
References
476
Index
481
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Preface
Symmetry pervades many forms of art and science, and group theory provides a systematic
way of thinking about symmetry. The mathematical concept of a group was invented in
1823 by E´variste Galois. Its applications in physical science developed rapidly during the
twentieth century, and today it is considered as an indispensable aid in many branches of
physics and chemistry. This book provides a thorough introduction to the subject and could
form the basis of two successive one-semester courses at the advanced undergraduate and
graduate levels. Some features not usually found in an introductory text are detailed
discussions of induced representations, the Dirac characters, the rotation group, projective
representations, space groups, magnetic crystals, and spinor bases. New concepts or
applications are illustrated by worked examples and there are a number of exercises.
Answers to exercises are given at the end of each section. Problems appear at the end of
each chapter, but solutions to problems are not included, as that would preclude their use as
problem assignments. No previous knowledge of group theory is necessary, but it is
assumed that readers will have an elementary knowledge of calculus and linear algebra
and will have had a first course in quantum mechanics. An advanced knowledge of
chemistry is not assumed; diagrams are given of all molecules that might be unfamiliar
to a physicist.
The book falls naturally into two parts. Chapters 1–10 (with the exception of a few
marked sections) are elementary and could form the basis of a one-semester advanced
undergraduate course. This material has been used as the basis of such a course at the
University of Western Ontario for many years and, though offered as a chemistry course, it
was taken also by some physicists and applied mathematicians. Chapters 11–18 are at a
necessarily higher level; this material is suited to a one-semester graduate course.
Throughout, explanations of new concepts and developments are detailed and, for the
most part, complete. In a few instances complete proofs have been omitted and detailed
references to other sources substituted. It has not been my intention to give a complete
bibliography, but essential references to core work in group theory have been given. Other
references supply the sources of experimental data and references where further development of a particular topic may be followed up.
I am considerably indebted to Professor Boris Zapol who not only drew all the diagrams
but also read the entire manuscript and made many useful comments. I thank him also for
his translation of the line from Alexander Pushkin quoted below. I am also indebted to my
colleague Professor Alan Allnatt for his comments on Chapters 15 and 16 and for several
fruitful discussions. I am indebted to Dr. Peter Neumann and Dr. Gabrielle Stoy of Oxford
xi
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xii
Preface
University for their comments on the proof (in Chapter 12) that multiplication of quaternions is associative. I also thank Richard Jacobs and Professor Amy Mullin for advice on
computing.
Grateful acknowledgement is made to the following for permission to make use of
previously published material:
The Chemical Society of Japan, for Figure 10.3;
Taylor and Francis Ltd ( for Table 10.2;
Cambridge University Press for Figure 12.5;
The American Physical Society and Dr. C. J. Bradley for Table 14.6.
‘‘Qprfelye krg le qeonhq preqy . . . ’’
A. Q. Orwihl
‘‘19 miq~ap~’’
which might be translated as:
‘‘Who serves the muses should keep away from fuss,’’ or, more prosaically,
‘‘Life interferes with Art.’’
I am greatly indebted to my wife Mary Mullin for shielding me effectively from
the daily intrusions of ‘‘Life’’ and thus enabling me to concentrate on this particular
work of ‘‘Art.’’
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Notation and conventions
General mathematical notation
)
P
8
iff
9
a
Cn
c, s
c2, s2
cx
cm
n
i
q1 q 2 q 3
<3
sx
T(n)
identically equal to
leads logically to; thus p ) q means if p is true, then q follows
P
sum of (no special notation is used when
is applied to sets, since it will
P
always be clear from the context when
means a sum of sets)
all
if and only if
there exists
the negative of a (but note ¼ Â in Chapter 13 and R ¼ ER, an operator in the
double group G, in Chapter 8)
n-dimensional space in which the components of vectors are complex
numbers
cos , sin
cos 2 , sin 2
x cos
cosmp=nị
p
imaginary unit, defined by i2 ẳ 1
quaternion units
n-dimensional space, in which the components of vectors are real numbers
configuration space, that is the three-dimensional space of real vectors in which
symmetry operations are represented
x sin
tensor of rank n in Section 15.1
Sets and groups
{gi}
2
62
A!B
a!b
the set of objects gi, i ¼ 1, . . . , g, which are generally referred to as ‘elements’
belongs to, as in gi 2 G
does not belong to
map of set A onto set B
map of element a (the pre-image of b) onto element b (the image of a)
xiii
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xiv
Notation and conventions
A\B
A[B
G
E, or g1
g
H, A, B
H&G
A$B
C
ck
C kij
intersection of A and B, that is the set of all the elements that belong to both A
and B
the union of A and B, that is the set of all the elements that belong to A, or to B,
or to both A and B
a group G ¼ {gi}, the elements gi of which have specific properties
(Section 1.1)
the identity element in G
the order of G, that is the number of elements in G
groups of order h, a, and b, respectively, often subgroups of G
H is a subset of G; if {hi} have the group properties, H is a subgroup of G
of order h
the groups A and B are isomorphous
a cyclic group of order c
the class of gk in G (Section 1.2) of order ck
Nc
P
class constants in the expansion i j ¼
Cijk k (Section A2.2)
k¼1
gi(ck)
G
K
li
ls
lv
Nc
Nrc
Nr
Ns
Nv
N(H|G)
t
ith element of the kth class
a group consisting of a unitary subgroup H and the coset AH, where A is an
antiunitary operator (Section 13.2), such that G ¼ {H} È A{H}
the kernel of G, of order k (Section 1.6)
dimension of ith irreducible representation
dimension of an irreducible spinor representation
dimension of an irreducible vector representation
number of classes in G
number of regular classes
number of irreducible representations
number of irreducible spinor representations
number of irreducible vector representations
the normalizer of H in G, of order n (Section 1.7)
t
P
index of a coset expansion of G on H, G ¼
gr H, with gr 62 H except for
r¼1
g1 ¼ E; {gr} is the set of coset representatives in the coset expansion of G, and
k, (ck)
{gr} is not used for G itself.
the centralizer of hj in G, of order z (Section 1.7)
an abbreviation for Z(gi|G)
ck
P
Dirac character of ck, equal to
gi (ck)
AB
A£ B
A^B
AB
AB
(outer) direct product of A and B, often abbreviated to DP
inner direct product of A and B
semidirect product of A and B
symmetric direct product of A and B (Section 5.3)
antisymmetric direct product of A and B (Section 5.3)
Z(hj|G)
Zi
i¼1
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Notation and conventions
xv
Vectors and matrices
a polar vector (often just a vector) which changes sign under inversion; r
may be represented by the directed line segment OP, where O is the origin
of the coordinate system
coordinates of the point P and therefore the components of a vector r ¼ OP;
independent variables in the function f (x, y, z).
space-fixed right-handed orthonormal axes, collinear with OX, OY, OZ
unit vectors, initially coincident with x y z, but firmly embedded in
configuration space (see R( n) below). Note that {e1 e2 e3} behave like
polar vectors under rotation but are invariant under inversion and
therefore they are pseudovectors. Since, in configuration space the vector
r ẳ e1x ỵ e2y ỵ e3z changes sign on inversion, the components of r, {x y z},
must change sign on inversion and are therefore pseudoscalars
unit vectors in a space of n dimensions, i ¼ 1, . . . , n
P
components of the vector v ¼
ei vi
r
xyz
xyz
e1 e2 e3
{ei}
{vi}
i
A
Ars, ars
En
det A or |ars|
AB
Cpr,qs
A[ij]
ha1 a2 . . . an|
ha|
|b1 b2 . . . bni
ha0 |
he|ri
the matrix A ¼ [ars], with m rows and n columns so that r ¼ 1, . . . , m
and s ¼ 1, . . . , n. See Table A1.1 for definitions of some special matrices
element of matrix A common to the rth row and sth column
unit matrix of dimensions n  n, in which all the elements are zero except
those on the principal diagonal, which are all unity; often abbreviated to E
when the dimensions of E may be understood from the context
determinant of the square matrix A
direct product of the matrices A and B
element apqbrs in C ¼ AB
ijth element (which is itself a matrix) of the supermatrix A
a matrix of one row containing the set of elements {ai}
an abbreviation for ha1 a2 . . . an|. The set of elements {ai} may be basis
vectors, for example he1 e2 e3|, or basis functions h 1 2 . . . n|.
a matrix of one column containing the set of elements {bi}, often
abbreviated to |bi; hb| is the transpose of |bi
the transform of ha| under some stated operation
an abbreviation for the matrix representative of a vector r; often given fully
as he1 e2 e3 | x y zi
Brackets
h|,|i
Dirac bra and ket, respectively; no special notation is used to distinguish the
bra and ket from row and column matrices, since which objects are intended
will always be clear from the context
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xvi
Notation and conventions
[A, B]
[a, A]
[a ; A]
[gi ; gj]
[n1 n2 n3]
commutator of A and B equal to AB À BA
complex number a ỵ iA
quaternion (Chapter 11)
projective factor, or multiplier (Chapter 12); often abbreviated to
[i ; j]
components of the unit vector n, usually given without the normalization
factor; for example, [1 1 1] are the components of the unit vector that makes
equal angles with OX, OY, OZ, the normalization factor 3À½ being
understood. Normalization factors will, however, be given explicitly when
they enter into a calculation, as, for example, in calculations using
quaternions
Angular momenta
L, S, J
^ ^
^
L
, S, J
L, S, J
^j
j ẳ j1 ỵ j2
orbital, spin, and total angular momenta
quantum mechanical operators corresponding to L, S, and J
quantum numbers that quantize L2, S2, and J2
operator that obeys the angular momentum commutation relations
total (j) and individual (j1, j2, . . . ) angular momenta, when angular momenta
are coupled
Symmetry operators and their matrix representatives
A
E
E
I
I1 I2 I3
I^1 I^2 I^3
I3
I
In
J x Jy Jz
antiunitary operator (Section 13.1); A, B may also denote linear,
Hermitian operators according to context
identity operator
operator R(2p n) introduced in the formation of the double group
G ¼ fR Rg from G ¼ {R}, where R ¼ ER (Section 8.1)
inversion operator
operators that generate infinitesimal rotations about x y z, respectively
(Chapter 11)
function operators that correspond to I1 I2 I3
matrix representative of I3, and similarly (note that the usual symbol
À(R) for the matrix representative of symmetry operator R is not used in
this context, for brevity)
generator of infinitesimal rotations about n, with components I1, I2, I3
matrix representative of In ¼nÁ I
matrix representatives of the angular momentum operators J^x , J^y , J^z
for the basis hm| ¼ h1=2 , À1=2|. Without the numerical factors of ½, these
are the Pauli matrices s1 s2 s3
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Notation and conventions
R( n)
^
Rð
nÞ
R, S, T
s^x s^y s^z
T
U
u
À(R)
À(R)pq
À
À 1 % À2
À¼
P
c i Ài
i
À ' Ài
P
Ài ¼
ci, j Àj
j
Àij ¼
P
k
n
1 2 3
Â
cij, k Àk
xvii
rotation through an angle about an axis which is the unit vector n;
here n is not a product but a single symbol n that fixes the three
independent parameters necessary to describe a rotation (the three
components of n, [n1 n2 n3], being connected by the normalization
condition); however, a space is inserted between and n in rotation
operators for greater clarity, as in R(2p=3 n). The range of
is Àp < p. R acts on configuration space and on all vectors therein
(including {e1 e2 e3}) (but not on {x y z}, which define the space-fixed
axes in the active representation)
function operator that corresponds to the symmetry operator R( n),
^ rị ẳ f R1 rị (Section 3.5)
defined so that Rf
general symbols for point symmetry operators (point symmetry
operators leave at least one point invariant)
spin operators whose matrix representatives are the Pauli matrices
and therefore equal to J^x , J^y , J^z without the common factor
of 1=2
translation operator (the distinction between T a translation operator
and T when used as a point symmetry operator will always be clear from
the context)
a unitary operator
time-evolution operator (Section 13.1)
matrix representative of the symmetry operator R; sometimes just R,
for brevity
pqth element of the matrix representative of the symmetry operator R
matrix representation
the matrix representations À1 and À2 are equivalent, that is related by a
similarity transformation (Section 4.2)
the representation À is a direct sum of irreducible representations Ài,
and each Ài occurs ci times in the direct sum À; when specific
representations (for example T1u) are involved, this would be written
c(T1u)
the reducible representation À includes Ài
the representation Ài is a direct sum of irreducible representations
Àj and each Àj occurs ci, j times in the direct sum Ài
Clebsch–Gordan decomposition of the direct product
Àij ¼ Ài £ Àj ; cij, k are the Clebsch–Gordan coefficients
reflection in the plane normal to n
the Pauli matrices (Section 11.6)
time-reversal operator
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xviii
Notation and conventions
Bases
he1 e2 e3|
hRx Ry Rz|
hu jÀj . . . u jj j
hu jm j
hu |
hu0 0 |
| u i
| UÀ1 U0 U1i
N
basis consisting of the three unit vectors {e1 e2 e3} initially coincident with
{x y z} but embedded in a unit sphere in configuration space so that
Rhe1 e2 e3 j ¼ he10 e20 e30 j ¼ he1 e2 e3j ÀðRÞ. The 3 Â 3 matrix À(R) is the
matrix representative of the symmetry operator R. Note that he1 e2 e3| is
often abbreviated to he|. If r 2 <3, R r ẳ Rhejri ẳ he0 jri ẳ hejRịjri ẳ
hejr0 i, which shows that he| and |ri are dual bases, that is they are
transformed by the same matrix À(R)
basis comprising the three infinitesimal rotations Rx, Ry, Rz about OX, OY,
OZ respectively (Section 4.6)
basis consisting of the 2j ỵ 1 functions, umj , Àj m j, which are
eigenfunctions of the z component of the angular momentum operator J^z ,
and of J^2 , with the Condon and Shortley choice of phase. The angular
momentum quantum numbers j and m may be either an integer or a halfinteger. For integral j the u jm are the spherical harmonics
Ylm ð ’Þ; ym
l ð ’Þ are the spherical harmonics written without
normalization factors, for brevity
an abbreviation for hu jÀj . . . u jj j, also abbreviated to hm|
spinor basis, an abbreviation for huẵ
ẵ j ẳ hjẵ ½i j½ À½ij, or h½ À½j in
the hm| notation
transform of hu | in C2, equal to hu |A
dual of hu |, such that |u0 0 i ¼ A| u i
matrix representation of the spherical vector U 2 C3 which is the dual of
0 1
the basis hyÀ1
1 y1 y1 j
normalization factor
Crystals
an ¼ ha| ni
bm ¼ hb| mi
lattice translation vector; an ¼ ha1 a2 a3| n1 n2 n3i (Section 16.1) (n is often
used as an abbreviation for the an)
reciprocal lattice vector; bm ¼ hb1 b2 b3| m1 m2 m3i ¼ he1 e2 e3| mx my mzi
(Section 16.3); m is often used as an abbreviation for the components of bm
Abbreviations
1-D
AO
BB
bcc
CC
CF
one-dimensional (etc.)
atomic orbital
bilateral binary
body-centered cubic
complex conjugate
crystal field
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Notation and conventions
CG
CR
CS
CT
DP
fcc
FE
FT
hcp
HSP
IR
ITC
L, R
LA
LCAO
LI
LO
LS
LVS
MO
MR
N
ORR
OT
PBC
PF
PR
RS
RS
sc
SP
TA
TO
ZOA
xix
Clebsch–Gordan
commutation relation
Condon and Shortley
charge transfer
direct product
face-centered cubic
free electron
fundamental theorem
hexagonal close-packed
Hermitian scalar product
irreducible representation
International Tables for Crystallography (Hahn (1983))
left and right, respectively, as in L and R cosets
longitudinal acoustic
linear combination of atomic orbitals
linearly independent
longitudinal optic
left- side (of an equation)
linear vector space
molecular orbital
matrix representative
north, as in N pole
Onsager reciprocal relation
orthogonality theorem
periodic boundary conditions
projective factor
projective representation
right side (of an equation)
Russell–Saunders, as in RS coupling or RS states
simple cubic
scalar product
transverse acoustic
transverse optic
zero overlap approximation
Cross-references
The author (date) system is used to identify a book or article in the list of references, which
precedes the index.
Equations in a different section to that in which they appear are referred to by
eq. (n1 Á n2 Á n3), where n1 is the chapter number, n2 is the section number, and n3 is the
equation number within that section. Equations occurring within the same section are referred
to simply by (n3). Equations are numbered on the right, as usual, and, when appropriate,
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xx
Notation and conventions
a number (or numbers) on the left, in parentheses, indicates that these equations are used in the
derivation of that equation so numbered. This convention means that such phrases as ‘‘it
follows from’’ or ‘‘substituting eq. (n4) in eq. (n5)’’ can largely be dispensed with.
Examples and Exercises are referenced, for example, as Exercise n1 Á n2-n3, even within
the same section. Figures and Tables are numbered n1 Á n3 throughout each chapter. When a
Table or Figure is referenced on the left side of an equation, their titles are abbreviated to T
or F respectively, as in F16.1, for example.
Problems appear at the end of each chapter, and a particular problem may be referred to
as Problem n1 Á n3, where n1 is the number of the chapter in which Problem n3 is to be found.
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1 The elementary properties
of groups
1.1
Definitions
All crystals and most molecules possess symmetry, which can be exploited to simplify the
discussion of their physical properties. Changes from one configuration to an indistinguishable configuration are brought about by sets of symmetry operators, which form particular
mathematical structures called groups. We thus commence our study of group theory with
some definitions and properties of groups of abstract elements. All such definitions and
properties then automatically apply to all sets that possess the properties of a group,
including symmetry groups.
Binary composition in a set of abstract elements {gi}, whatever its nature, is always
written as a multiplication and is usually referred to as ‘‘multiplication’’ whatever it
actually may be. For example, if gi and gj are operators then the product gi gj means
‘‘carry out the operation implied by gj and then that implied by gi.’’ If gi and gj are both
n-dimensional square matrices then gi gj is the matrix product of the two matrices gi and gj
evaluated using the usual row  column law of matrix multiplication. (The properties of
matrices that are made use of in this book are reviewed in Appendix A1.) Binary
composition is unique but is not necessarily commutative: gi gj may or may not be equal
to gj gi. In order for a set of abstract elements {gi} to be a G, the law of binary composition
must be defined and the set must possess the following four properties.
(i) Closure. For all gi, with gj 2 {gj},
gi g j ¼ g k 2 fg i g,
gk a unique element of fgi g:
(1)
Because gk is a unique element of {gi}, if each element of {gi} is multiplied from the left,
or from the right, by a particular element gj of {gi} then the set {gi} is regenerated with the
elements (in general) re-ordered. This result is called the rearrangement theorem
gj fgi g ¼ fgi g ¼ fgi g gj :
(2)
Note that {gi} means a set of elements of which gi is a typical member, but in no
particular order. The easiest way of keeping a record of the binary products of the
elements of a group is to set up a multiplication table in which the entry at the
intersection of the gith row and gjth column is the binary product gi gi ¼ gk, as in
Table 1.1. It follows from the rearrangement theorem that each row and each column of
the multiplication table contains each element of G once and once only.
1
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2
The elementary properties of groups
Table 1.1. Multiplication table for the group G ¼ {gi} in which the product
gi gj happens to be gk.
G
gi
gj
gk
gi
gj
gk
..
.
gi2
gj gi
gk gi
gk
g2j
gk gj
gi gk
gj gk
gk2
...
(ii) Multiplication is associative. For all gi, gj, gk 2 {gi},
gi ðg j gk ị ẳ gi g j ịgk :
(3)
(iii) The set {gi} contains the identity element E, with the property
E g j ¼ gj E ¼ gj , 8 gj 2 fgi g:
(4)
(iv) Each element gi of {gi} has an inverse g À1
i 2 fggi such that
À1
gÀ1
gi ¼ gi g À1
i
i ¼ E, g i 2 fg i g, 8 g i 2 fg i g:
(5)
The number of elements g in G is called the order of the group. Thus
G ¼ fgi g,
i ¼ 1, 2, . . . , g:
(6)
When this is necessary, the order of G will be displayed in parentheses G(g), as in G(4) to
indicate a group of order 4.
Exercise 1.1-1 With binary composition defined to be addition: (a) Does the set of
positive integers {p} form a group? (b) Do the positive integers p, including zero (0)
form a group? (c) Do the positive (p) and negative (Àp) integers, including zero, form a
group? [Hint: Consider the properties (i)–(iv) above that must be satisfied for {gi} to form
a group.]
The multiplication of group elements is not necessarily commutative, but if
gi gj ¼ gj gi , 8 gi , gj 2 G
(7)
then the group G is said to be Abelian. Two groups that have the same multiplication table
are said to be isomorphous. As we shall see, a number of other important properties of a
group follow from its multiplication table. Consequently these properties are the same for
isomorphous groups; generally it will be necessary to identify corresponding elements in
the two groups that are isomorphous, in order to make use of the isomorphous property. A
group G is finite if the number g of its elements is a finite number. Otherwise the group G is
infinite, if the number of elements is denumerable, or it is continuous. The group of
Exercise 1.1-1(c) is infinite. For finite groups, property (iv) is automatically fulfilled as
a consequence of the other three.
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1.1 Definitions
3
If the sequence gi , g 2i , g3i , . . . starts to repeat itself at gcỵ1
ẳ gi , because gci ¼ E, then
i
the set fgi g2i g 3i . . . gci ¼ Eg, which is the period of gi, is a group called a cyclic group,
C. The order of the cyclic group C is c.
Exercise 1.1-2 (a) Show that cyclic groups are Abelian. (b) Show that for a finite
cyclic group the existence of the inverse of each element is guaranteed. (c) Show that
! ¼ exp(À2pi=n) generates a cyclic group of order n, when binary composition is
defined to be the multiplication of complex numbers.
If every element of G can be expressed as a finite product of powers of the elements in a
particular subset of G, then the elements of this subset are called the group generators. The
choice of generators is not unique: generally, a minimal set is employed and the defining
relations like gi ¼ (gj) p (gk)q, etc., where {gj gk} are group generators, are stated. For
example, cyclic groups are generated from just one element gi.
Example 1.1-1 A permutation group is a group in which the elements are permutation
operators. A permutation operator P rearranges a set of indistinguishable objects. For example, if
Pfa b c . . .g ¼ fb a c . . .g
(8)
then P is a particular permutation operator which interchanges the objects a and b. Since
{a b . . .} is a set of indistinguishable objects (for example, electrons), the final configuration {b a c . . . } is indistinguishable from the initial configuration {a b c . . . } and P is a
particular kind of symmetry operator. The best way to evaluate products of permutation
operators is to write down the original configuration, thinking of the n indistinguishable
objects as allocated to n boxes, each of which contains a single object only. Then write
down in successive rows the results of the successive permutations, bearing in mind that a
permutation other than the identity involves the replacement of the contents of two or more
boxes. Thus, if P applied to the initial configuration means ‘‘interchange the contents of
boxes i and j’’ (which initially contain the objects i and j, respectively) then P applied to
some subsequent configuration means ‘‘interchange the contents of boxes i and j, whatever
they currently happen to be.’’ A number of examples are given in Table 1.2, and these
should suffice to show how the multiplication table in Table 1.3 is derived. The reader
should check some of the entries in the multiplication table (see Exercise 1.1-3).
The elements of the set {P0 P1 . . . P5} are the permutation operators, and binary
composition of two members of the set, say P3 P5, means ‘‘carry out the permutation
specified by P5 and then that specified by P3.’’ For example, P1 states ‘‘replace the contents
of box 1 by that of box 3, the contents of box 2 by that of box 1, and the contents of box 3 by
that of box 2.’’ So when applying P1 to the configuration {3 1 2}, which resulted from P1 (in
order to find the result of applying P21 ¼ P1 P1 to the initial configuration) the contents of
box 1 (currently 3) are replaced by those of box 3 (which happens currently to be 2 – see the
line labeled P1); the contents of box 2 are replaced by those of box 1 (that is, 3); and finally
the contents of box 3 (currently 2) are replaced by those of box 2 (that is, 1). The resulting
configuration {2 3 1} is the same as that derived from the original configuration {1 2 3} by
P2, and so
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