**Instructor:** Hemanshu Kaul

**Office:** 125C, Engineering 1

**Phone:** (312) 567-3128

**E-mail:** kaul [at] iit.edu

**Time:** 11:25am, Tuesday & Thursday.

**Place:** 106, Stuart Bldg.

**Office Hours:** 3:15-4:15pm Tuesday and Thursday, and by appointment.

Emailed questions are also encouraged.

**Graduate Teaching Assistant:** Hansen Ha; hha3 [at] hawk.iit.edu.

**TA Office Hours:** 1pm-3pm Tuesday, 129, E1 Building.

|Course Information| |Advice| |Announcements| |Examinations| |Homework| |Class Log & Handouts| |Info Links|

The

The author's views on the value of Abstract algebra.

The official course syllabi.

Author's advice for learning Abstract Algebra,

and his advice on learning proofs. Required reading.

Excellent advice by Doug West on how to write homework solutions in a course like this. Required reading.

Why do we have to learn proofs?

Understanding Mathematics - a study guide

On a more abstract note, here is a discussion of Language and Grammar of Mathematics - which is what you are starting to learn in a course like this.

Excellent advice for math majors, especially those planning to go on to graduate school, by Terry Tao, 2006 Fields medallist. Required reading.

*Thursday, 9/6*: Dates and Syllabi for Exam #1 and Exam #2 and the Final Exam have been announced below.*Thursday, 8/26*: HW#1 has been uploaded below. Also see the accompanying comments file.*Tuesday, 8/24*: Check this webpage regularly for homework assignments, announcements, etc.

*Exam # 1*: Thursday October 4th. Topics: All the topics (covered in class and through textbook) corresponding to HW#1 to HW#5.*Exam # 2*: Thursday, November 8th. Topics: All the topics (covered in class and through textbook) corresponding to HW#6 to HW#9.*Final Exam*: December 3rd, Monday, 10:30am to 12:30pm. Topics: All topics studied during the semester.

Words like "construct", "show", "obtain", "determine", etc., typically mean that proof is required. Full credit to most problems requires proof of statements made. Use sentences; you cannot give a proof without words. Results covered in class can be used without proof if you state/refer to them correctly.

Practice Problems: If you have time, think about these problems for extra practice.

Written Problems: You have to submit written solution to all these problems.

All problem numbers below are based on the

*Thursday, 8/23*:**Read**Examples 4 and 5, and UPC scheme on pages 7-10; Equivalence relations and their examples on pages 17-20; Functions and their examples on pages 20-23.**Homework #1 :**Due Thursday, 8/30. Solutions distributed in class on 8/30, Thursday.

Practice Problems: Chapter 0 - #5, #7, #9, #15, #20-21, #33, #57, #59.

Written Problems: Chapter 0 - #8, #12, #14, #18, #26, #34, #52, #58.

*Thursday, 8/30*:**Read**examples 1-21 and table 2.1 on pages 43-49.**Homework #2 :**Due Thursday, 9/6. Solutions distributed in class on 9/6, Thursday.

Practice Problems: Chapter 1: #8, #10, #17. Chapter 2: #8, #9, #15, #26, #49, #53.

Written Problems: Chapter 1: #1&2, #5, #14, #16. Chapter 2: #14, #16, #18, #19, #22, #52, #54.

*Thursday, 9/6*:**Read**examples 1-8 on pages 60-64.**Homework #3 :**Due Thursday, 9/13. Solutions distributed in class on 9/13, Thursday.

Practice Problems: Chapter 2: #23, #33, #36, #47. Chapter 3: #1, #5, #6, #11, #14, #19, #26.

Written Problems: Chapter 2: #20, #32, #34, #51. Chapter 3: #4, #7, #8, #18, #52.

*Thursday, 9/13*:**Read**examples 9-15 and theorems 3.5 and 3.6 on pages 65-68.**Homework #4 :**Due Thursday, 9/20. Solutions distributed in class on 9/20, Thursday.

Practice Problems: Chapter 3: #10, #13, #21, #22, #23, #33, #38, #43, #51, #53, #67, #68, #70, #71, #75, #79.

Written Problems: Chapter 3: #14, #16, #32, #34, #39, #42, #50, #63, #77.

*Thursday, 9/20*:**Read**examples 1-4 on pages77-78 and theorem 4.3, examples 5-6 on pages 82-84.**Homework #5:**Due Thursday, 9/27. Solutions distributed in class on 9/27, Thursday.

Practice Problems: Chapter 4: #5, #8, #16, #17, #21, #31, #42, #58.

Written Problems: Chapter 4: #14, #22, #24, #28, #30, #38, #40, #52, #82.

*Thursday, 9/27*:**Read**examples on page 82 following Cor 4 on pg.81, and on page 83 following Theorem 4.3, Examples on page 84 and Theorem/Cor on page 85. Examples 1-3 on pages 100-101.*Thursday, 10/4*:**Read**examples 4-8 on pages 106-112.**Homework #6:**Due Thursday, 10/11. Solutions distributed in class on 10/18, Thursday.

Practice Problems: Chapter 5: #2, #5, #6, #8, #9, #10, #11, #15, #19, #25, #35, #36, #45, #47, #52, #57, #59, #61, #71, #72, #73.

Written Problems: Chapter 5: #14, #16, #23 and 24, #26, #32, #40, #46, #50, #54.

*Thursday, 10/11*:**Read**examples 1-7 on pages 129-130, pay close attention to examples 5 and 7.**Read**the statements of Theorems 6.1 and 6.2.**Read**example 9 on page 135.**Homework #7:**Due Thursday, 10/18. Solutions distributed in class on 10/18, Thursday.

Practice Problems: Chapter 6: #1, #3, #4, 35, #7, #20, #25, #27, #34.

Written Problems: Chapter 6: #6, #8, #10, #22, #28, #36.

*Thursday, 10/18*:**Read**example 8 on page 132, Complete statement of Theorem 6.3 and discussion following it on page 134, examples 10-13 on pages 135-137.**Homework #8:**Due Thursday, 10/25. Solutions distributed in class on 11/1, Thursday.

Practice Problems: Chapter 6: #9, #11, #15, #18, #37, #41, #42, #45, #54.

Written Problems: Chapter 6: #12, #16, #30, #38, #50, #52, #58.

*Thursday, 10/25*:**Read**examples 1-6 on pages 144-150, Definition of Orbit of a point and complete statement of Theorem 7.4.**Homework #9:**Due Thursday, 11/1. Solutions distributed in class on 11/6, Tuesday.

Practice Problems: Chapter 7: #1, 3, 10, 11, 15, 23, 25, 27, 28, 33, 37, 42, 45, 49, 52, 53.

Written Problems: Chapter 7: #8, #12, #20, #26, #34, #38, one of{#40 or #46}, #50.

Comment: It might be useful to apply Theorem 4.4 and its corollary to solve some of these problems, please review them.

*Thursday, 11/1*:**Read**examples 4, 5, 6 on pages 164-165.**Read**examples 5-9 and 10-12 on pages 186-189.**Homework #10:**Due Thursday, 11/15. Solutions distributed in class on 11/15, Thursday.

Practice Problems: Chapter 8: #2, 3, 4, 5, 9, 13, 14, 19, 21, 23, 25, 27, 30-32, 39, 49, 52, 54, 55, 58, 61. Chapter 9: #1, 6-8, 11-13, 16, 17, 28, 37, 39, 43, 47.

Written Problems: Chapter 8: #6, #8, #12, #24, #26, #34, #44, #60. Chapter 9: #2, #9, #40.

*Thursday, 11/15*:**Read**examples 1-7 on pages 208-209.**Read**examples 9, 10, 12, 13 on pages 213-215.*Tuesday, 11/20*:**Read**examples 1-7 on pages 246-247.**Read**examples 1-8 on pages 255-256.**Read**the discussion on pages 256-259 (to be covered on Tues 11/27).**Homework #11:**UPDATED! Due Thursday, 11/29. Solutions distributed in class on 11/29, Thursday.

Practice Problems: Chapter 9: #48, 56, 59, 71, 72. Chapter 10: #3, 4, 5, 9, 16, 18, 22, 35, 40, 41, 44, 48. Chapter 12: #1, 2, 3, 6, 8, 17, 18, 19, 21, 24, 25, 31, 33, 39, 41, 49. Chapter 13: #2, 4, 5, 11, 15, 18, 30, 35, 39, 49, 51, 57, 67.

Written Problems: Chapter 9: #50, 72. Chapter 10: #8, #10, #14, #30, #42. Chapter 12: #4, 30, 42, 50. Chapter 13: #8, #12, #28, #38.

*Tuesday, 8/21*: Divisor, Prime numbers, Division Algorithm, Well Ordering Principle and its application to the proof of correctness of the Division Algo (existence and uniqueness). (From Chapter 0)*Thursday, 8/23*: Handouts with Course Information and discussion of Course requirements and aim, Advice for doing HW, `What this course is really about'. gcd and lcm, their properties, Two methods for computing gcd, Correctness of Euclidean Algorithm, Integer linear combinations, GCD as integer linear combination, Euclid's Lemma and its proof, Fundamental Thm of Arithmetic and prime factorization, Modular Arithmetic, First Principle of Mathematical Induction. (From Chapter 0)*Tuesday, 8/28*: Proof of GCD as smallest linear combination; Symmetries in the plane, Symmetries of a square - rotational and reflective, Composition of symmetries, Cayley table, Abelian and non-Abelian operations, Dihedral Groups, Rotational groups. (From Chapters 0 and 1)*Thursday, 8/30*: Definition of binary operations and of Group - examples and non-examples including R, Z_n, GL(n,R), etc. (From Chapter 2)*Tuesday, 9/4*: Extended Euclid's lemma w. proof, applied to show existence of inverses in U(n) - U(n) is a group, Proof for uniqueness of group identity, Cancellation law for groups. (From Chapter 2)*Thursday, 9/6*: Cancelation law, uniqueness of inverses, Inverse of product, order of a group and order of an element with examples, Subgroup and observations about its definition, examples using Cayley tables, One-step Subgroup test and how it implies the definition of subgroup, examples. (From Chapters 2 and 3)*Tuesday, 9/11*: Two-step subgroup test - application to subgroup of elements of order at most 2 and subgroup of the form HK where H and K are given subgroups, Finite subgroup test and application to showing cyclic subgroup, Order of an element is at most the order of the group, U_k(n) - subgroups or not of U(n). (From Chapter 3)*Thursday, 9/13*: U_k(n) - subgroups or not of U(n) - theorem and hints how to prove it, Cyclic subgroup generated by an element - examples, Center of a group, Centralizer of an element; Quiz#1. (From Chapter 3)*Tuesday, 9/18*: Center and centralizer are subgroups. examples of both using Cayley tables, Center of D_n, Proof of when U_n(n) is a subgroup of U(n), Theorem for is a^i=a^j and |a|=||, if a^k=e then |a| | k.(From Chapters 3 and 4)*Thursday, 9/20*: Discussion of Quiz#1 problems. Statements, proofs of criteria for a^i=a^j and the related corollaries relating order of element with order of subgroup, etc., order of power of an element and cyclic group generated by a power of an element, and corollaries regarding generators of finite cyclic groups, and generators of Z_n. (From Chapter 4)*Tuesday, 9/25*: Examples for order of power of an element and cyclic group generated by a power of an element, Fundamental Theorem of cyclic groups with examples and part of the proof, Permutations as functions and two-line notation. (From Chapters 4 and 5)*Thursday, 9/27*: Examples of permutations, their composition and inverses, S_n group of permutations of n symbols and why it is a group, One-line notation and teh cycle notation for permutations, Every permutation can be written as a prodcut of disjoint cycles w. algo, Elements of D_4 as permutations and its generating set, Disjoint cycles commute, Order of a permutation in S_n. (From Chapter 5) *Tuesday, 10/2*: Order of a permutation in S_n, Writing permutations as product of 2-cycles, Even and odd permutations, Identity permutation is always even, A_n the subgroup of all even permutations, Order of A_n by a bijection, Rotations of tetrahedron and how to represent them as permutations. (From Chapter 5)*Thursday, 10/4*: Exam #1.*Tuesday, 10/9*: Isomorphism - motivation, definitions and examples, non-examples and how to show two groups are not isomorphic, isomorphism class of infinite cyclic group and finite cyclic group, isomorphism through Cayley tables, Automorphism and an automorphism of positive reals under multiplication, Cayley's theorem. (From Chapter 6)*Thursday, 10/11*: Discussion of Exam #1 solutions and grades.*Tuesday, 10/16*: Proof of Cayley's theorem, Properties of isomorphisms - identity, powers, commutativity, cyclic generators, order - with proofs. (From Chapter 6)*Thursday, 10/18*: Properties of isomorphisms - order, solutions of an equation, elements of same order - with proofs, Properties of Isomorphisms acting on groups, Inner automorphisms, Aut(G) and Inn(G) are groups, Aut(Z_n) is isomorphic to U(n), illustration for n=8. (From Chapter 6)*Tuesday, 10/23*: How image of 1 determines the whole automorphism in Z_n, proof of Aut(Z_n) is isomorphic to U(n), Left and right cosets, fundamental properties of cosets, cosets form a partition of equal size blocks, proof of Lagrange's Theorem and its corollaries. (From Chapters 6 and 7)*Thursday, 10/25*: Converse of Lagrange's theorem is not true, HK - when its a group and what is its order, Classification of all groups of order 2p, Stabilizer of a point under a permutation with an example. (From Chapter 7)*Tuesday, 10/30*: Orbit and the orbit-stabilizer theorem with examples, External Direct Product of Groups, Classification of groups of order 4, Order of elements under direct product, when is a direct product cyclic, Direct product decomposition of Z_n, U(n) as direct product. (From Chapters 7 and 8)*Thursday, 11/1*: the orbit-stabilizer theorem with examples, U(n) as direct product of Z_n, Normal subgroups and their characterization, examples of normal subgroups and its relation to center and Abelian properties. (From Chapters 8 and 9)*Tuesday, 11/6*: Product of a normal subgroup with another subgroup is always a subgroup, A_n is a normal subgroup of S_n, Factor group of left cosets of a normal subgroup, Factor group of Z with nZ is isomorphic to Z, Index 2 subgroups are normal and using it to show A_n has no subgroup of order 6, IF G/Z(G) is cyclic then G is Abelian. (From Chapter 9)*Thursday, 11/8*: Exam #2.*Tuesday, 11/13*: The G/Z theorem, its proof and application to non-Abelian group of order pq, Cauchy's theorem for Abelian groups, Homomorphisms and comparison to Isomorphisms, Kernel of a homomorphism, Examples and non-examples, Properties of homomorphisms. (From Chapters 9 and 10)*Thursday, 11/15*: Properties of subgroups under homomorphisms, Kernels are normal subgroups, The fundamental Isomorphisms theorem, Statement of Fundamental Theorem of Abelian Groups; Discussion of grades and solutions of Exam #2. (From Chapter 10 and 11)*Tuesday, 11/20*: Completion of solutions of Exam#2. Rings - examples and non-examples, Commutative Rings, Unity, Units, Properties of Rings, Subring and test for subrings, Zero divisor, Integral Domain - examples and non-examples, Cancelation law for ID. (From Chapters 12 and 13)*Tuesday, 11/27*: Field - two definitions, Every finite ID is field, examples of finite fields, q(sqrt(2)) is a field, Characteristic of a ring - examples, Relation between the char(R) and additive order of unity, Char(Field) = 0 or p. (From Chapters 12 and 13)*Thursday, 11/29*: Rings of polynomials over a ring, If R is ID then R[x] is also ID, Division algorithm for F[x] where F is a field, consequences of the DA in F[x] to roots and factorization of polynomials in F[x]. (From Chapter 13 and 16)

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