Algebra 2. Linear Algebra, Galois Theory, Representation theory, Group extensions and Schur Multiplier

435 Pages·2017·3.17 MB·English

by  Ramji Lal

Ramji Lal
Algebra 2
Linear Algebra, Galois Theory,
Representation Theory,
Group Extensions and Schur Multiplier
123
Ramji Lal
HarishChandra Research Institute (HRI)
Allahabad, Uttar Pradesh
India
ISSN 2363-6149 ISSN 2363-6157 (electronic)
Infosys Science FoundationSeries
ISSN 2364-4036 ISSN 2364-4044 (electronic)
Infosys Science FoundationSeries in MathematicalSciences
ISBN978-981-10-4255-3 ISBN978-981-10-4256-0 (eBook)
DOI 10.1007/978-981-10-4256-0
LibraryofCongressControlNumber:2017935547
©SpringerNatureSingaporePteLtd.2017
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerNatureSingaporePteLtd.
Theregisteredcompanyaddressis:
152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore
Preface
Algebra has played a central and decisive role in all branches of mathematics and,
inturn,inallbranchesofscienceandengineering.Itisnotpossibleforalecturerto
cover, physically in a classroom, the amount of algebra which a graduate student
(irrespective of the branch of science, engineering, or mathematics in which he
preferstospecialize)needstomaster.Inaddition,thereareavarietyofstudentsina
class. Some of them grasp the material very fast and do not need much of assis-
tance. At the same time, there are serious students who can do equally well by
putting a little more effort. They need some more illustrations and also more
exercisestodeveloptheirskillandconfidenceinthesubjectbysolvingproblemson
theirown.Again,itisnotpossibleforalecturertodosufficientlymanyillustrations
and exercises in the classroom for the aforesaid purpose. This is one of the con-
siderations which prompted me to write a series of three volumes on the subject
starting from the undergraduate level to the advance postgraduate level. Each
volume is sufficiently rich with illustrations and examples together with numerous
exercises. These volumes also cater for the need of the talented students with
difficult, challenging, and motivating exercises which were responsible for the
furtherdevelopmentsinmathematics.Occasionally,theexercisesdemonstratingthe
applications in different disciplines are also included. The books may also act as a
guidetoteachersgivingthecourses.Theresearchers workinginthefieldmayalso
find it useful.
The first volume consists of 11 chapters, which starts with language of mathe-
matics(logicandsettheory)andcentersaround theintroductiontobasicalgebraic
structures, viz., groups, rings, polynomial rings, and fields together with funda-
mentalsinarithmetic.Thisvolumeservesasabasictextforthefirst-yearcoursein
algebra at the undergraduate level. Since this is the first introduction to the
abstract-algebraic structures, we proceed rather leisurely in this volume as com-
pared with the other volumes.
The present (second) volume contains 10 chapters which includes the funda-
mentals of linear algebra, structure theory of fields and the Galois theory, repre-
sentationtheoryofgroups,andthetheoryofgroupextensions.Itisneedlesstosay
that linear algebra is the most applicable branch of mathematics, and it is essential
for students of any discipline to develop expertise in the same. As such, linear
algebraisanintegralpartofthesyllabusattheundergraduatelevel.Indeed,avery
significant andessentialpart (Chaps. 1–5)of linear algebracovered inthis volume
doesnotrequireanybackgroundmaterialfromVolume1ofthebookexceptsome
amount of set theory. General linear algebra over rings, Galois theory, represen-
tation theory of groups, and the theory of group extensions follow linear algebra,
andindeedthesearepartsofthesyllabusforthesecond-andthethird-yearstudents
ofmostoftheuniversities.Assuch,thisvolumetogetherwiththefirstvolumemay
serve as a basic text for the first-, second-, and third-year courses in algebra.
The third volume of the book contains 10 chapters, and it can act as a text for
graduateand advance graduate studentsspecializinginmathematics.This includes
commutative algebra, basics in algebraic geometry, semi-simple Lie algebras,
advancerepresentationtheory,andChevalleygroups.Thetableofcontentsgivesan
idea of the subject matter covered in the book.
There is no prerequisite essential for the book except, occasionally, in some
illustrationsandexercises,someamountofcalculus,geometry,ortopologymaybe
needed. An attempt to follow the logical ordering has been made throughout
the book.
My teacher (Late) Prof. B.L. Sharma, my colleague at the University of
Allahabad, my friend Dr. H.S. Tripathi, my students Prof. R.P. Shukla, Prof.
Shivdatt,Dr.BrajeshKumarSharma,Mr.SwapnilSrivastava,Dr.AkhileshYadav,
Dr. Vivek Jain, Dr. Vipul Kakkar, and above all, the mathematics students of the
University of Allahabad had always been the motivating force for me to write a
book. Without their continuous insistence, it would have not come in the present
form. I wish to express my warmest thanks to all of them.
Harish-Chandra Research Institute (HRI), Allahabad, has always been a great
sourceformetolearnmoreandmoremathematics.Iwishtoexpressmydeepsense
of appreciation and thanks to HRI for providing me all infrastructural facilities to
write these volumes.
Lastbutnotleast,IwishtoexpressmythankstomywifeVeenaSrivastavawho
had always been helpful in this endeavor.
Inspiteofallcare,somemistakesandmisprintsmighthavecreptinandescaped
myattention.Ishallbegratefultoanysuchattention.Criticismsandsuggestionsfor
the improvement of the book will be appreciated and gratefully acknowledged.
Allahabad, India Ramji Lal
April 2017
Contents
1 Vector Spaces ... .... .... ..... .... .... .... .... .... ..... .. 1
1.1 Concept of a Field... ..... .... .... .... .... .... ..... .. 1
1.2 Concept of a Vector Space (Linear Space).. .... .... ..... .. 7
1.3 Subspaces.. .... .... ..... .... .... .... .... .... ..... .. 11
1.4 Basis and Dimension. ..... .... .... .... .... .... ..... .. 16
1.5 Direct Sum of Vector Spaces, Quotient of a Vector Space .. .. 23
2 Matrices and Linear Equations.. .... .... .... .... .... ..... .. 31
2.1 Matrices and Their Algebra. .... .... .... .... .... ..... .. 31
2.2 Types of Matrices ... ..... .... .... .... .... .... ..... .. 35
2.3 System of Linear Equations. .... .... .... .... .... ..... .. 40
2.4 Gauss Elimination, Elementary Operations, Rank,
and Nullity. .... .... ..... .... .... .... .... .... ..... .. 43
2.5 LU Factorization .... ..... .... .... .... .... .... ..... .. 58
2.6 Equivalence of Matrices, Normal Form.... .... .... ..... .. 60
2.7 Congruent Reduction of Symmetric Matrices.... .... ..... .. 65
3 Linear Transformations... ..... .... .... .... .... .... ..... .. 73
3.1 Definition and Examples ... .... .... .... .... .... ..... .. 73
3.2 Isomorphism Theorems .... .... .... .... .... .... ..... .. 75
3.3 Space of Linear Transformations, Dual Spaces .. .... ..... .. 79
3.4 Rank and Nullity.... ..... .... .... .... .... .... ..... .. 83
3.5 Matrix Representations of Linear Transformations.... ..... .. 85
3.6 Effect of Change of Bases on Matrix Representation.. ..... .. 88
4 Inner Product Spaces. .... ..... .... .... .... .... .... ..... .. 97
4.1 Definition, Examples, and Basic Properties . .... .... ..... .. 97
4.2 Gram–Schmidt Process .... .... .... .... .... .... ..... .. 107
4.3 Orthogonal Projection, Shortest Distance... .... .... ..... .. 112
4.4 Isometries and Rigid Motions ... .... .... .... .... ..... .. 120
5 Determinants and Forms .. ..... .... .... .... .... .... ..... .. 131
5.1 Determinant of a Matrix.... .... .... .... .... .... ..... .. 131
5.2 Permutations ... .... ..... .... .... .... .... .... ..... .. 135
5.3 Alternating Forms, Determinant of an Endomorphism. ..... .. 139
5.4 Invariant Subspaces, Eigenvalues. .... .... .... .... ..... .. 150
5.5 Spectral Theorem, and Orthogonal Reduction ... .... ..... .. 159
5.6 Bilinear and Quadratic Forms ... .... .... .... .... ..... .. 176
6 Canonical Forms, Jordan and Rational Forms.. .... .... ..... .. 195
6.1 Concept of a Module over a Ring .... .... .... .... ..... .. 195
6.2 Modules over P.I.D .. ..... .... .... .... .... .... ..... .. 203
6.3 Rational and Jordan Forms . .... .... .... .... .... ..... .. 214
7 General Linear Algebra... ..... .... .... .... .... .... ..... .. 229
7.1 Noetherian Rings and Modules .. .... .... .... .... ..... .. 229
7.2 Free, Projective, and Injective Modules.... .... .... ..... .. 234
7.3 Tensor Product and Exterior Power... .... .... .... ..... .. 250
7.4 Lower K-theory. .... ..... .... .... .... .... .... ..... .. 258
8 Field Theory, Galois Theory .... .... .... .... .... .... ..... .. 265
8.1 Field Extensions. .... ..... .... .... .... .... .... ..... .. 265
8.2 Galois Extensions.... ..... .... .... .... .... .... ..... .. 275
8.3 Splitting Field, Normal Extensions.... .... .... .... ..... .. 284
8.4 Separable Extensions. ..... .... .... .... .... .... ..... .. 294
8.5 Fundamental Theorem of Galois Theory ... .... .... ..... .. 305
8.6 Cyclotomic Extensions..... .... .... .... .... .... ..... .. 311
8.7 Geometric Constructions ... .... .... .... .... .... ..... .. 318
8.8 Galois Theory of Equation.. .... .... .... .... .... ..... .. 324
9 Representation Theory of Finite Groups... .... .... .... ..... .. 331
9.1 Semi-simple Rings and Modules . .... .... .... .... ..... .. 331
9.2 Representations and Group Algebras.. .... .... .... ..... .. 346
9.3 Characters, Orthogonality Relations... .... .... .... ..... .. 351
9.4 Induced Representations.... .... .... .... .... .... ..... .. 361
10 Group Extensions and Schur Multiplier... .... .... .... ..... .. 367
10.1 Schreier Group Extensions.. .... .... .... .... .... ..... .. 368
10.2 Obstructions and Extensions .... .... .... .... .... ..... .. 391
10.3 Central Extensions, Schur Multiplier .. .... .... .... ..... .. 398
10.4 Lower K-Theory Revisited.. .... .... .... .... .... ..... .. 418
Bibliography.... .... .... .... ..... .... .... .... .... .... ..... .. 427
Index.. .... .... .... .... .... ..... .... .... .... .... .... ..... .. 429
Notation Algebra 1
hai Cyclic subgroup generated by a, p. 122
a/b a divides b, p. 57
a * b a is an associate of b, p. 57
At The transpose of a matrix A, p. 200
AH The hermitian conjugate of a matrix A, p. 215
Aut(G) The automorphism group of G, p. 105
A The alternating group of degree n, p. 175
n
Bðn;RÞ Borel subgroup, p. 187
C ðHÞ The centralizer of H in G, p. 159
G
C The field of complex numbers, p. 78
D The dihedral group of order 2n, p. 90
n
det Determinant map, p. 191
End(G) Semigroup of endomorphisms of G, p. 105
f(A) Image of A under the map f, p. 34
f−1(B) Inverse image of B under the map f, p. 34
f|Y Restriction of the map f to Y, p. 30
E‚ Transvections, p. 200
ij
Fit(G) Fitting subgroup, p. 353
g.c.d. Greatest common divisor, p. 58
g.l.b. Greatest lower bound, or inf, p. 40
G=lHðG=rHÞ The set of left(right) cosets of G mod H, p. 135
G/H The quotient group of G modulo H, p. 151
½G:H(cid:2) The index of H in G, p. 135
jGj Order of G, p. 331
G0 ¼ ½G; G(cid:2) Commutator subgroup of G, p. 403
Gn nth term of the derived series of G, p. 345
GLðn;RÞ General linear group, p. 186
I Identity map on X, p. 30
X
i Inclusion map from Y, p. 30
Y
Inn(G) The group of inner automorphisms, p. 407
ker f The kernel of the map f, p. 35
L ðGÞ nth term of the lower central series of G, p. 281
n
l.c.m. Least common multiple, p. 58
l.u.b. Least upper bound, or sup, p. 40
M (R) The ring of n(cid:3)n matrices with entries in R, p. 350
n
N Natural number system, p. 21
N ðHÞ Normalizer of H in G, p. 159
G
O(n) Orthogonal group, p. 197
O(1, n) Lorentz orthogonal group, p. 201
PSO(1, n) Positive special Lorentz orthogonal group, p. 201
Q The field of rational numbers, p. 74
Q The quaternion group, p. 88
8
R The field of real numbers, p. 75
R(G) Radical of G, p. 346
S Symmetric group of degree n, p. 88
n
Sym(X) Symmetric group on X, p. 88
S3 The group of unit quaternions, p. 92
hSi Subgroup generated by a subset S, p. 116
SLðn;RÞ Special linear group, p. 196
SO(n) Special orthogonal group, p. 197
SO(1, n) Special Lorentz orthogonal group, p. 201
SPð2n;RÞ Symplectic group, p. 202
SU(n) Special unitary group, p. 202
U(n) Unitary group, p. 202
U Group of prime residue classes modulo m, p. 100
m
V Kleins four group, p. 102
4
X/R The quotient set of X modulo R, p. 36
R Equivalence class modulo R determined by x, p. 27
x
X+ Successor of X, p. 20
XY The set of maps from Y to X, p. 34
(cid:4) Proper subset, p. 14
}QðXÞ Power set of X, p. 19
n G Direct product of groups G ;1(cid:5)k(cid:5)n, p. 142
k¼1 k k
/ Normal subgroup, p. 147
// Subnormal subgroup, p. 332
Z(G) Center of G, p. 108
Z The ring of residue classes modulo m, p. 256
m
p(n) The number of partition of n, p. 172
H (cid:6)K Semidirect product of H with K, p. 204
pffiffiffi
A Radical of an ideal A, p. 286
R(G) Semigroup ring of a ring R over a semigroup G, p. 238
R[X] Polynomial ring over the ring R in one variable, p. 240
R½X ;X ;(cid:7)(cid:7)(cid:7);X (cid:2) Polynomial ring in several variables, p. 247
1 2 n
„ The Mobius function, p. 256
(cid:2)(cid:3) (cid:4) Sum of divisor function, p. 256
a Legendre symbol, p. 280
p
Stab(G, X) Stabilizer of an action of G on X, p. 295
G Isotropy subgroup of an action of G at x, p. 295
x
XG Fixed point set of an action of G on X, p. 296
Z (G) nth term of the upper central series of G, p. 351
n
ΦðGÞ The Frattini subgroup of G, p. 355
Notation Algebra 2
B2ðK;HÞ Group of 2 co-boundaries with given (cid:2), p. 385
(cid:2)
C(A) Column space of A, p. 42
Ch(G, K) Set of characters from G to K, p. 278
Ch(G) Character ring of G, p. 350
dim(V) Dimension of V, p. 18
EXT Category of Schreier group extensions, p. 368
E(H, K) The set of equivalence classes of extensions of H by K, p. 376
E ]E Baer sum of extensions, p. 388
1 2
EXTˆðH;KÞ Set of equivalence classes of extensions associated to abstract
kernel ˆ, p. 384
E(V) Exterior algebra of V, p. 257
FACS Category offactor systems, p. 375
F(X) The fixed field of a set of automorphism of a field, p. 275
G(L/K) The Galois group of the field extension L of K, p. 275
G^G Non-abelian exterior square of a group G, p. 413
(cid:2)
K Algebraic closure of K, p. 289
H2ðK;HÞ Second cohomology with given (cid:2), p. 385
(cid:2)
K ðRÞ Grothendieck group of the ring R, p. 257
0
K ðRÞ Whitehead group of the ring R, p. 260
1
KL Separable closure of K in L, p. 295
S
L/K Field extension L of K, p. 262
m ðXÞ Minimum polynomial of linear transformation T, p. 212
T
min ðfiÞðXÞ Minimum polynomial of fi over the field K, p. 265
K
M(V) Group of rigid motion on V, p. 122
M(cid:8) N Tensor product of R-modules M and N, p. 250
R
NL=K Norm map from L to K, p. 279
N(A) Null space of A, p. 41
ObsðˆÞ Obstruction of the abstract kernel ˆ, p. 393
R(A) Row space of A, p. 42
St(R) Steinberg group, p. 422