MAT4233

Course Catalog

MAT 4233. Abstract Algebra II. (3-0) 3 Credit Hours.

Prerequisites: MAT2233 and MAT3013. Basic properties and examples of semigroups, monoids, and groups, detailed study of permutation, dihedral, and congruence groups, cyclic groups, normal subgroups, quotient groups, homomorphism, isomorphism theorems, direct products of groups, The Sylow Theorems, rings and fields and their basic properties, ideals, polynomial rings. Generally offered: Spring. Differential Tuition: $150.

Description

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study. The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

Evaluation

• Two midterms (for classes that meet twice a week) and an optional final.

• Exam score is the best of final score and midterm average.

• Students will have access to several past exams for practice.

Text

J. Gallian, Contemporary abstract algebra (8e) Houghton Mifflin

Topics List A

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
1.0	0	The natural order on N and the well ordering principle Mathematical induction Construction of Z and its properties (graph the equivalence classes) Division algorithm Congruence mod m Algebra on the quotient set Z_m GCD, LCM, Bézout Primes, Euclid's Lemma Fundamental Theorem of Arithmetic nan	Sets Partitions Equivalence relations and classes Functions Images and preimages	Review of known facts about Z A concrete introduction to techniques of abstract algebra
2.0	2	Symmetries Properties of composition Definition of a group Elementary proofs with groups: uniqueness of identity uniqueness of inverses cancellation shortcuts to establishing group axioms*14: 4, 6, 10. Foundational examples with Cayley tables	Sets and functions	Motivation for the concept of a group Learn the definition of a group Learn basic automatic properties of groups (with proofs) for later use as shortcuts Starting to build a catalog of examples of groups Learn to construct and read Cayley tables
3.0	10, 6	Cayley's theorem Homomorphisms of groups Isomorphisms and their inverses Automorphisms Examples	Functions Groups Matrix multiplication Change of basis for matrices	General framework for thinking of groups as symmetries and motivation for homomorphisms Learn the definitions of homomorphism and isomorphism Prove that homomorphisms preserve powers. Starting to build a catalog of examples of homomorphisms.
4.0	3, 10	Definition of a subgroup Subgroup tests Automatic closure under inverses for finite subgroups Subgroups generated by a subset Examples Images and preimages under a homomorphism are subgroups. Fibers as cosets of the kernel First Isomorphism Theorem Examples	Groups Functions Equivalence relations and classes	Learn how to identify subgroups, with proofs. Learn how to obtain new groups from old via homomorphisms. Learn how to prove a homomorphism is one-to-one by using the kernel.
5.0		Euclidean space as an additive group Null space and column space of a linear map Solutions to linear inhomogeneous systems Invertible linear transformations and matrices, GL(n,R) Determinant: homomorphism, similarity invariance, geometrical interpretation. Additive and multiplicative subgroups of complex numbers
6.0	4	Order of a group, order of an element Defining homomorphisms on Z (free group) Classification of cyclic groups Subgroups of cyclic groups and their generators Subgroup lattice
7.0
8.0	5, 1	Cycle notation D_n as a subgroup of S_n Factoring into disjoint cycles Ruffini's theorem Cyclic subgroups, powers of a permutation Parity, A_n < S_n
9.0	7, 9	Cosets as equivalence classes Lagrange's theorem Fermat's little theorem Euler's theorem Normal subgroups Factor groups Universal property of factor groups First Isomorphism theorem revisited
10.0	8, 9, 11	External direct product Universal property of direct product Chinese Remainder Theorem Internal direct product Free product Universal property of free product Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
11.0	12, 13, 15, 16	Motivation and definition Properties Subrings Integral domains Fields Characteristic Ring homomorphisms Examples
12.0	14	Ideals Ideals generated by a set, principal ideals Images and preimages of ideals are ideals Factor rings Prime ideals Maximal ideals Localization, field of quotients
13.0	16, 17, 18	Division algorithm for F[x] F[x] is a PID Factorization of polynomials Fundamental Theorem of Algebra Tests, Eisenstein's criterion Irreducibles and associates Z[x] is a UFD
14.0
15.0

Topics List B

Week	Session	Topics	Chapter	Prerequisite Skills	Learning Outcomes	Examples	Exercises
1	Preliminaries	The natural order on N and the well ordering principle Mathematical induction Construction of Z and its properties (graph the equivalence classes) Division algorithm Congruence mod m Algebra on the quotient set Z_m GCD, LCM, Bézout Primes, Euclid's Lemma Fundamental Theorem of Arithmetic	0	Sets Partitions Equivalence relations and classes Functions Images and preimages	Review of known facts about Z A concrete introduction to techniques of abstract algebra	Equivalence classes are partitions If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f. Introduce congruence using the remainder function. Congruence classes mod 3 Extended Euclid's algorithm	0: 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.
2	Groups	Symmetries Properties of composition Definition of a group Elementary proofs with groups: uniqueness of identity uniqueness of inverses cancellation shortcuts to establishing group axioms*14: 4, 6, 10. Foundational examples with Cayley tables	2	Sets and functions	Motivation for the concept of a group Learn the definition of a group Learn basic automatic properties of groups (with proofs) for later use as shortcuts Starting to build a catalog of examples of groups Learn to construct and read Cayley tables	Z, Q, Q*, Q+, R, R*, R+, {-1, 1} R^n, M(n,R) symmetric group S_n Z_2 defined for now as {even,odd} ({solids,stripes}) correspondence of Z_2 with {-1, 1} and with S_2 functions X -> G with pointwise operation (fg)(x)=f(x)g(x) free group on a finite set	2: 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.
3	Homomorphisms	Cayley's theorem Homomorphisms of groups Isomorphisms and their inverses Automorphisms Examples	10, 6	Functions Groups Matrix multiplication Change of basis for matrices	General framework for thinking of groups as symmetries and motivation for homomorphisms Learn the definitions of homomorphism and isomorphism Prove that homomorphisms preserve powers. Starting to build a catalog of examples of homomorphisms.	R -> R: x -> ax R^n -> R^n: v -> Av M(n,R) -> M(n,R): X -> AX R* -> R*: x -> x^n R -> R+: x -> a^x (a>0) determinant: GL(n,R) -> R* inclusions natural projection Z -> Z_2 evaluation {X -> G} -> G: f -> f(a) Z -> Z: k -> -k Aut(Z_2) is trivial Aut(Z_3) is isomorphic to Z_2 change of basis S in R^n gives an inner automorphism of GL(n,R): X -> S^(-1).X.S C -> C: z -> complex conjugate of z	10: 1, 2, 3, 6, 15, 21, 24, 25. 6: 3, 8, 14, 17, 35, 39, 49, 58, 61.
4	Subgroups	Definition of a subgroup Subgroup tests Automatic closure under inverses for finite subgroups Subgroups generated by a subset Examples Images and preimages under a homomorphism are subgroups. Fibers as cosets of the kernel First Isomorphism Theorem Examples	3, 10	Groups Functions Equivalence relations and classes	Learn how to identify subgroups, with proofs. Learn how to obtain new groups from old via homomorphisms. Learn how to prove a homomorphism is one-to-one by using the kernel.	Cyclic subgroups <x>={x^k: k in Z} or xZ={xk: k in Z}	3: 4, 7, 11, 28, 29, 32.
5	Groups in Linear Algebra and Complex Variable	Euclidean space as an additive group Null space and column space of a linear map Solutions to linear inhomogeneous systems Invertible linear transformations and matrices, GL(n,R) Determinant: homomorphism, similarity invariance, geometrical interpretation. Additive and multiplicative subgroups of complex numbers				GL(n,R), O(n,R), SL(n,R), SO(n,R) C, C*, S^1, n-th roots of unity, D_n as a subgroup of O(2,R^2) Möbius transformations on the Riemann sphere
6	Cyclic groups	Order of a group, order of an element Defining homomorphisms on Z (free group) Classification of cyclic groups Subgroups of cyclic groups and their generators Subgroup lattice	4				4: 10, 14, 18, 32, 52.
7	Catch up and review Midterm 1
8	Permutations	Cycle notation D_n as a subgroup of S_n Factoring into disjoint cycles Ruffini's theorem Cyclic subgroups, powers of a permutation Parity, A_n < S_n	5, 1				5: 3, 5, 11, 21, 22, 24, 26, 32, 36, 38. 1: 1, 2, 13, 22.
9	Cosets	Cosets as equivalence classes Lagrange's theorem Fermat's little theorem Euler's theorem Normal subgroups Factor groups Universal property of factor groups First Isomorphism theorem revisited	7, 9			cosets of <(1,2)> in S_3 cosets of a flip in D_4 inverse images of subgroups are normal, kernels A_n is normal in S_n rotations in D_n Z/nZ R/Z	7: 17, 20, 24, 26, 33, 43, 60. 9: 12, 13, 14.
10	Products	External direct product Universal property of direct product Chinese Remainder Theorem Internal direct product Free product Universal property of free product Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)	8, 9, 11			ZxZ public key cryptography free product of Z with itself in groups and in Abelian groups free group on a set free Abelian group on a set	8: 2, 5, 12, 54, 55, 56. 9: 22, 24, 28, 31, 32, 48, 49, 50. 11: 3, 4.
11	Rings	Motivation and definition Properties Subrings Integral domains Fields Characteristic Ring homomorphisms Examples	12, 13, 15, 16			Z and other number systems R*, Z_n* = U(n) polynomial rings	12: 4, 6, 7, 12, 13, 22, 26. 13: 7, 12. 15: 20, 8, 12, 13, 19.
12	Ideals and factor rings	Ideals Ideals generated by a set, principal ideals Images and preimages of ideals are ideals Factor rings Prime ideals Maximal ideals Localization, field of quotients	14			mZ < Z <2, x> = 2Z[x]+xZ[x] < Z[x] Hausdorff Maximality Principle Q[x]/<x^2-2> R[x]/<x^2+1> Z -> Q polynomials -> rational functions	14: 4, 6, 8, 10, 28, 31.
13	Factorization	Division algorithm for F[x] F[x] is a PID Factorization of polynomials Fundamental Theorem of Algebra Tests, Eisenstein's criterion Irreducibles and associates Z[x] is a UFD	16, 17, 18			In Z[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))	16: 19, 20, 26, 32, 34. 17: 2, 4, 6, 11, 13, 14, 15, 16.
14	Catch up and review Midterm 2
15	Catch up and review for final Study days

See also

• UTSA Undergraduate Mathematics Course Descriptions