iv
Over the years I used the five chapters that were typed as a base for my algebra
courses, supplementing them as I saw fit. In 1996 I wrote a sixth chapter, giving
enough material for a full first year graduate course. This chapter was written in the
same “style” as the previous chapters, i.e., everything was right down to the nub. It
hung together pretty well except for the last two sections on determinants and dual
spaces. These were independent topics stuck on at the end. In the academic year
1997-98 I revised all six chapters and had them typed in LaTeX. This is the personal
background of how this book came about.
It is difficult to do anything in life without help from friends, and many of my
friends have contributed to this text. My sincere gratitude goes especially to Marilyn
Gonzalez, Lourdes Robles, Marta Alpar, John Zweibel, Dmitry Gokhman, Brian
Coomes, Huseyin Kocak, and Shulim Kaliman. To these and all who contributed,
this book is fondly dedicated.
This book is a survey of abstract algebra with emphasis on linear algebra. It is
intended for students in mathematics, computer science, and the physical sciences.
The first three or four chapters can stand alone as a one semester course in abstract
algebra. However they are structured to provide the background for the chapter on
linear algebra. Chapter 2 is the most difficult part of the book because groups are
written in additive and multiplicative notation, and the concept of coset is confusing
at first. After Chapter 2 the book gets easier as you go along. Indeed, after the
first four chapters, the linear algebra follows easily. Finishing the chapter on linear
algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6
continues the material to complete a first year graduate course. Classes with little
background can do the first three chapters in the first semester, and chapters 4 and 5
in the second semester. More advanced classes can do four chapters the first semester
and chapters 5 and 6 the second semester. As bare as the first four chapters are, you
still have to truck right along to finish them in one semester.
The presentation is compact and tightly organized, but still somewhat informal.
The proofs of many of the elementary theorems are omitted. These proofs are to
be provided by the professor in class or assigned as homework exercises. There is a
non-trivial theorem stated without proof in Chapter 4, namely the determinant of the
product is the product of the determinants. For the proper flow of the course, this
theorem should be assumed there without proof. The proof is contained in Chapter 6.
The Jordan form should not be considered part of Chapter 5. It is stated there only
as a reference for undergraduate courses. Finally, Chapter 6 is not written primarily
for reference, but as an additional chapter for more advanced courses.
v
This text is written with the conviction that it is more effective to teach abstract
and linear algebra as one coherent discipline rather than as two separate ones. Teach-
ing abstract algebra and linear algebra as distinct courses results in a loss of synergy
and a loss of momentum. Also with this text the professor does not extract the course
from the text, but rather builds the course upon it. I am convinced it is easier to
build a course from a base than to extract it from a big book. Because after you
extract it, you still have to build it. The bare bones nature of this book adds to its
flexibility, because you can build whatever course you want around it. Basic algebra
is a subject of incredible elegance and utility, but it requires a lot of organization.
This book is my attempt at that organization. Every effort has been extended to
make the subject move rapidly and to make the flow from one topic to the next as
seamless as possible. The student has limited time during the semester for serious
study, and this time should be allocated with care. The professor picks which topics
to assign for serious study and which ones to “wave arms at”. The goal is to stay
focused and go forward, because mathematics is learned in hindsight. I would have
made the book shorter, but I did not have any more time.
When using this text, the student already has the outline of the next lecture, and
each assignment should include the study of the next few pages. Study forward, not
just back. A few minutes of preparation does wonders to leverage classroom learning,
and this book is intended to be used in that manner. The purpose of class is to
learn, not to do transcription work. When students come to class cold and spend
the period taking notes, they participate little and learn little. This leads to a dead
class and also to the bad psychology of “O K, I am here, so teach me the subject.”
Mathematics is not taught, it is learned, and many students never learn how to learn.
Professors should give more direction in that regard.
Unfortunately mathematics is a difficult and heavy subject. The style and
approach of this book is to make it a little lighter. This book works best when
viewed lightly and read as a story. I hope the students and professors who try it,
enjoy it.
E. H. Connell
Department of Mathematics
University of Miami
Coral Gables, FL 33124
ec@math.miami.edu
vi
Outline
Chapter 1 Background and Fundamentals of Mathematics
Sets, Cartesian products 1
Relations, partial orderings, Hausdorff maximality principle, 3
equivalence relations
Functions, bijections, strips, solutions of equations, 5
right and left inverses, projections
Notation for the logic of mathematics 13
Integers, subgroups, unique factorization 14
Chapter 2 Groups
Groups, scalar multiplication for additive groups 19
Subgroups, order, cosets 21
Normal subgroups, quotient groups, the integers mod n 25
Homomorphisms 27
Permutations, the symmetric groups 31
Product of groups 34
Chapter 3 Rings
Rings 37
Units, domains, fields 38
The integers mod n 40
Ideals and quotient rings 41
Homomorphisms 42
Polynomial rings 45
Product of rings 49
The Chinese remainder theorem 50
Characteristic 50
Boolean rings 51
Chapter 4 Matrices and Matrix Rings
Addition and multiplication of matrices, invertible matrices 53
Transpose 56
Triangular, diagonal, and scalar matrices 56
Elementary operations and elementary matrices 57
Systems of equations 59
vii
Determinants, the classical adjoint 60
Similarity, trace, and characteristic polynomial 64
Chapter 5 Linear Algebra
Modules, submodules 68
Homomorphisms 69
Homomorphisms on R
n
71
Cosets and quotient modules 74
Products and coproducts 75
Summands 77
Independence, generating sets, and free basis 78
Characterization of free modules 79
Uniqueness of dimension 82
Change of basis 83
Vector spaces, square matrices over fields, rank of a matrix 85
Geometric interpretation of determinant 90
Linear functions approximate differentiable functions locally 91
The transpose principle 92
Nilpotent homomorphisms 93
Eigenvalues, characteristic roots 95
Jordan canonical form 96
Inner product spaces, Gram-Schmidt orthonormalization 98
Orthogonal matrices, the orthogonal group 102
Diagonalization of symmetric matrices 103
Chapter 6 Appendix
The Chinese remainder theorem 108
Prime and maximal ideals and UFD
s
109
Splitting short exact sequences 114
Euclidean domains 116
Jordan blocks 122
Jordan canonical form 123
Determinants 128
Dual spaces 130
viii
1 2 3 4
5 6 7 8
9 11 10
Abstract algebra is not only a major subject of science, but it is also
magic and fun. Abstract algebra is not all work and no play, and it is
certainly not a dull boy. See, for example, the neat card trick on page
18. This trick is based, not on sleight of hand, but rather on a theorem
in abstract algebra. Anyone can do it, but to understand it you need
some group theory. And before beginning the course, you might first try
your skills on the famous (some would say infamous) tile puzzle. In this
puzzle, a frame has 12 spaces, the first 11 with numbered tiles and the
last vacant. The last two tiles are out of order. Is it possible to slide the
tiles around to get them all in order, and end again with the last space
vacant? After giving up on this, you can study permutation groups and
learn the answer!
Chapter 1
Background and Fundamentals of
Mathematics
This chapter is fundamental, not just for algebra, but for all fields related to mathe-
matics. The basic concepts are products of sets, partial orderings, equivalence rela-
tions, functions, and the integers. An equivalence relation on a set A is shown to be
simply a partition of A into disjoint subsets. There is an emphasis on the concept
of function, and the properties of surjective, injective, and bijective. The notion of a
solution of an equation is central in mathematics, and most properties of functions
can be stated in terms of solutions of equations. In elementary courses the section
on the Hausdorff Maximality Principle should be ignored. The final section gives a
proof of the unique factorization theorem for the integers.
Notation Mathematics has its own universally accepted shorthand. The symbol
∃ means “there exists” and ∃! means “there exists a unique”. The symbol ∀ means
“for each” and ⇒ means “implies”. Some sets (or collections) are so basic they have
their own proprietary symbols. Five of these are listed below.
N = Z
+
= the set of positive integers = {1, 2, 3, }
Z = the ring of integers = { , −2, −1, 0, 1, 2, }
Q = the field of rational numbers = {a/b : a, b ∈ Z, b = 0}
R = the field of real numbers
C = the field of complex numbers = {a + bi : a, b ∈ R} (i
2
= −1)
Sets Suppose A, B, C, are sets. We use the standard notation for intersection
and union.
A ∩B = {x : x ∈ A and x ∈ B} = the set of all x which are elements
1
2 Background Chapter 1
of A and B.
A ∪B = {x : x ∈ A or x ∈ B} = the set of all x which are elements of
A or B.
Any set called an index set is assumed to be non-void. Suppose T is an index set and
for each t ∈ T , A
t
is a set.
t∈T
A
t
= {x : ∃ t ∈ T with x ∈ A
t
}
t∈T
A
t
= {x : if t ∈ T, x ∈ A
t
} = {x : ∀t ∈ T, x ∈ A
t
}
Let ∅ be the null set. If A ∩ B = ∅, then A and B are said to be disjoint.
Definition Suppose each of A and B is a set. The statement that A is a subset
of B (A ⊂ B) means that if a is an element of A, then a is an element of B. That
is, a ∈ A ⇒ a ∈ B. If A ⊂ B we may say A is contained in B, or B contains A.
Exercise Suppose each of A and B is a set. The statement that A is not a subset
of B means .
Theorem (De Morgan’s laws) Suppose S is a set. If C ⊂ S (i.e., if C is a subset
of S), let C
, the complement of C in S, be defined by C
= S −C = {x ∈ S : x ∈ C}.
Then for any A, B ⊂ S,
(A ∩B)
= A
∪ B
and
(A ∪B)
= A
∩ B
Cartesian Products If X and Y are sets, X ×Y = {(x, y) : x ∈ X and y ∈ Y }.
In other words, the Cartesian product of X and Y is defined to be the set of all
ordered pairs whose first term is in X and whose second term is in Y .
Example R × R = R
2
= the plane.
Chapter 1 Background 3
Definition If each of X
1
, , X
n
is a set, X
1
× ··· × X
n
= {(x
1
, , x
n
) : x
i
∈ X
i
for 1 ≤ i ≤ n} = the set of all ordered n-tuples whose i-th term is in X
i
.
Example R × ···× R = R
n
= real n-space.
Question Is (R ×R
2
) = (R
2
× R) = R
3
?
Relations
If A is a non-void set, a non-void subset R ⊂ A × A is called a relation on A. If
(a, b) ∈ R we say that a is related to b, and we write this fact by the expression a ∼ b.
Here are several properties which a relation may possess.
1) If a ∈ A, then a ∼ a. (reflexive)
2) If a ∼ b, then b ∼ a. (symmetric)
2
) If a ∼ b and b ∼ a, then a = b. (anti-symmetric)
3) If a ∼ b and b ∼ c, then a ∼ c. (transitive)
Definition A relation which satisfies 1), 2
), and 3) is called a partial ordering.
In this case we write a ∼ b as a ≤ b. Then
1) If a ∈ A, then a ≤ a.
2
) If a ≤ b and b ≤ a, then a = b.
3) If a ≤ b and b ≤ c, then a ≤ c.
Definition A linear ordering is a partial ordering with the additional property
that, if a, b ∈ A, then a ≤ b or b ≤ a.
Example A = R with the ordinary ordering, is a linear ordering.
Example A = all subsets of R
2
, with a ≤ b defined by a ⊂ b, is a partial ordering.
Hausdorff Maximality Principle (HMP) Suppose S is a non-void subset of A
and ∼ is a relation on A. This defines a relation on S. If the relation satisfies any
of the properties 1), 2), 2
), or 3) on A, the relation also satisfies these properties
when restricted to S. In particular, a partial ordering on A defines a partial ordering
4 Background Chapter 1
on S. However the ordering may be linear on S but not linear on A. The HMP is
that any linearly ordered subset of a partially ordered set is contained in a maximal
linearly ordered subset.
Exercise Define a relation on A = R
2
by (a, b) ∼ (c, d) provided a ≤ c and
b ≤ d. Show this is a partial ordering which is linear on S = {(a, a) : a < 0}. Find
at least two maximal linearly ordered subsets of R
2
which contain S.
One of the most useful applications of the HMP is to obtain maximal monotonic
collections of subsets.
Definition A collection of sets is said to be monotonic if, given any two sets of
the collection, one is contained in the other.
Corollary to HMP Suppose X is a non-void set and A is some non-void
collection of subsets of X, and S is a subcollection of A which is monotonic. Then ∃
a maximal monotonic subcollection of A which contains S.
Proof Define a partial ordering on A by V ≤ W iff V ⊂ W, and apply HMP.
The HMP is used twice in this book. First, to show that infinitely generated
vector spaces have free bases, and second, in the Appendix, to show that rings have
maximal ideals (see pages 87 and 109). In each of these applications, the maximal
monotonic subcollection will have a maximal element. In elementary courses, these
results may be assumed, and thus the HMP may be ignored.
Equivalence Relations A relation satisfying properties 1), 2), and 3) is called
an equivalence relation.
Exercise Define a relation on A = Z by n ∼ m iff n − m is a multiple of 3.
Show this is an equivalence relation.
Definition If ∼ is an equivalence relation on A and a ∈ A, we define the equiva-
lence class containing a by cl(a) = {x ∈ A : a ∼ x}.
Chapter 1 Background 5
Theorem
1) If b ∈ cl(a) then cl(b) = cl(a). Thus we may speak of a subset of A
being an equivalence class with no mention of any element contained
in it.
2) If each of U, V ⊂ A is an equivalence class and U ∩ V = ∅, then
U = V .
3) Each element of A is an element of one and only one equivalence class.
Definition A partition of A is a collection of disjoint non-void subsets whose union
is A. In other words, a collection of non-void subsets of A is a partition of A provided
any a ∈ A is an element of one and only one subset of the collection. Note that if A
has an equivalence relation, the equivalence classes form a partition of A.
Theorem Suppose A is a non-void set with a partition. Define a relation on A by
a ∼ b iff a and b belong to the same subset of the partition. Then ∼ is an equivalence
relation, and the equivalence classes are just the subsets of the partition.
Summary There are two ways of viewing an equivalence relation — one is as a
relation on A satisfying 1), 2), and 3), and the other is as a partition of A into
disjoint subsets.
Exercise Define an equivalence relation on Z by n ∼ m iff n − m is a multiple
of 3. What are the equivalence classes?
Exercise Is there a relation on R satisfying 1), 2), 2
) and 3) ? That is, is there
an equivalence relation on R which is also a partial ordering?
Exercise Let H ⊂ R
2
be the line H = {(a, 2a) : a ∈ R}. Consider the collection
of all translates of H, i.e., all lines in the plane with slope 2. Find the equivalence
relation on R
2
defined by this partition of R
2
.
Functions
Just as there are two ways of viewing an equivalence relation, there are two ways
of defining a function. One is the “intuitive” definition, and the other is the “graph”
or “ordered pairs” definition. In either case, domain and range are inherent parts of
the definition. We use the “intuitive” definition because everyone thinks that way.
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