361 Honors Algebra III, Fall 1996
A. Goetz

Textbook:  Abstract Algebra by I. N. Herstein, Third Edition.

Syllabus

1. Elements of set theory.
Definition: set, subset, union, intersection, difference (relative
complement), empty set, universe, complement. Venn diagrams.
Associative, commutative and distributive laws for union and
intersection. De'Morgan laws. Symmetric difference and its properties
(to be used later as an example of a ring of characteristic 2).
(Chapter 1, Section 2)

2. Mappings.
Injective, surjective and bijective mappings. Composition of mappings.
The associative property of the composition of mappings.  (Chapter 2,
Section 3) Permutations of a set A. Identity mapping, inverse mapping.
Group properties of permutations (Chapter 2, Section 4) S_n . Cycles,
disjoint cycle decomposition of a permutation. Transpositions, and the
representation of a permutation as a product of transpositions. Even
and odd permutations. Representation of every even permutation as a
composition of 3-cycles.  (Chapter 3, Sections 1-3)

3. Rudiments of number theory.
Arithmetical properties of integers. The principle of mathematical induction, the well ordering principle. Division with remainder. Divisibility. The greatest common divisor of two numbers. Euclid's algorithm. Relatively prime numbers.
(Chapter 1, Sections 5-6)
Congruences modulo n. Definition properties. Addition and
multiplication of congruences. Cancellation laws: ab =  ac  (mod n) is
equivalent to b = c  (mod n), if (a, n)= 1, ab = ac  (mod an) is
equivalent to b = c  (mod n) for any a \ne 0. Solutions of a
 Diophantine equation. Solutions of a congruence ax b  (mod n), and of
 a system of congruences of first degree.  Equivalence relations and
equivalence classes. Equivalence classes modulo n. Arithmetic of Z_n.
Invertible elements and zero divisors in Z_n .  (The
textbook introduces congruences modulo n  as examples in
 Section 2.4 and deals with them through problems to this section)

4. Complex numbers.
Definition. Addition and multiplication, complex conjugate, modulus,
inverse. Polar form, argument, multiplication in polar form.
De'Moivre's theorem, n-th roots of a complex number. n-th root of
unity, primitive roots.  (Chapter 2, Section 7)


5. Group theory.
Definitions: Group, Abelian group, subgroup. Examples of groups:
Previously encountered: Z,  Z_n, Q,  R, C, rectangular matrices
with addition, Q^+, R^+, C-{0}, nonsingular square matrices,
S_n and various groups of transformations, the power set of a non
empty set with symmetric difference. Newly introduced: U_n, the
set of all invertible elements of Z_n with multiplication,
dihedral groups. The 4 group and the quaternion group (Chapter 1,
Section 1)

Some elementary properties of groups: uniqueness of identity element
and inverses. Cancellation law etc.  (Chapter 2, Section 2) Cyclic
groups. Order of a group, order of an element. Cyclic subgroups, cyclic
groups (Chapter 2. Section 3)

Right and left cosets.  Lagrange's theorem and some simple
consequences, e. g. every group of prime order is cyclic, for every a
in G, a^{o(G)}  = e. Index if a subgroup. Euler's and Fermat's theorems
(Chapter 2, Section 4)

Isomorphisms of groups.  Examples: Dihedral group of order 6 and S
do5(3),  Square 2x2 matrices of form \pmatrix{a&b\cr-b&a\cr} and
complex numbers and many others. Isomorphism of cyclic groups.
Automorphisms. Inner automorphisms. Cayley's theorem.

Homomorphisms of groups. Kernel of an homomorphisms. One to one
homomorphisms homomorphisms onto. Normal subgroups. Counter example for
transitivity of the normal subgroup relation. Every right cosets of a
subgroup is a left coset if and only if the subgroup is normal.
(Chapter 2, section 5)

Factor groups. The natural homomorphism G \to G/N. Examples of factor
groups. Order of the factor group. Cauchy's theorem for Abelian
groups.  (Chapter 2, section 6) The first second and third homomorphism
theorems, the correspondence theorem.  (Chapter 2, Section 7)

Direct products of groups (Chapter 2, Section 9) Representation of an
Abelian group as a direct product of p-groups.  (Chapter 2, Section 10)

Conjugacy. Conjugacy in S_n.  Proof that S_5 is a simple group.
Conjugacy classes and the class equation.  Sylow's theorem. (The proof
of the first theorem given in class.  Assigned homework (Problems 27
and 28 on p. 107) was leading to a proof of the second theorem.
Students had difficulty with these problems and I have to discuss them
in class).  (Chapter 2, Section 11) Definitions of a normal series,
solvable groups, composition series. Jordan-H lder theorem mentioned
but not proved. Illustrated on examples.  (not in the textbook)

6. Ring theory.
Definition of a ring, commutative ring, ring
with unity.  Zero divisors, invertible elements, domain, integral
domain, division ring, field. Examples:  Z, Z_n, Q,  R, C,
quaternions, square matrices with entries from the above, the power set
with symmetric difference and intersection.  Subrings. Necessary and
sufficient condition for a subring. A subring of a domain is a domain.
If 1 is the unity of a ring and belongs to a subring, then it is the
unity of the subring.  A division ring is a domain (homework).
Example:  2Z_{10} \subset Z_10.  (Chapter 4, Sections 1,2) Homomorphisms
of rings. kernels of a homomorphism. Right ideal, left ideals and
two-sided ideals. Factor rings. Homomorphism theorems and
correspondence theorem for rings. Principal ideals, maximal ideals.
(Chapter 4, Sections 3,4)

7. Polynomial rings.
Polynomials over commutative rings and over fields. Equality of
polynomials. Division algorithm for polynomials over fields. Greatest
common divisor of polynomials. Euclid algorithm. Relatively prime
polynomial, irreducible polynomials, factorization. Principal ideals
and maximal ideals of F[x]. Factorization. Euclidean rings.  (Chapter