MTH 331/431 Abstract Algebra I (Fall 2019):
 What is Abstract (or Modern) Algebra?
Algebra is defined to be the study of algebraic structures. Mathematicians study algebraic structures from a general point of view, compare different structures, and find relationships between them. In this course we will study elementary number theory, groups, rings, and fields. Find more information about the course in the syllabus.  Instructor: Dr. Sofya Chepushtanova, office SLC 410, email: sofya.chepushtanova@wilkes.edu.
 Class meetings: MWF 11:0011:50am, room SLC 411 and R 1:001:50pm, SLC 378.
 Office Hours: MWF 9:009:50am, MW 2:002:50am, or by appointment, room SLC 410.
 Textbook: Tom Judson's Abstract Algebra: Theory and Applications.
An electronic version of the book is freely available here.  Sage: an opensource program for doing mathematics and is the ideal companion to Tom Judson's textbook. Sage includes many mature and powerful opensource tools for mathematics, such as GAP, a system for computational discrete algebra (used for group theory). With a strength in number theory, Sage also has excellent support for rings and fields. We will be using Sage sometimes for exercises and assignments. More information about Sage is here and here.
 Reading:
 LaTeX: a markup language to typeset documents. It excels at making math and the overall layout beautiful.
You can use it to type your homework solutions. Read the tutorial here. You may find the following LaTeX cheatsheet useful.
Consider the following LaTeX template to start: download (it is the template of document On Proofs).
Schedule:
Week of 
Class Topics  Remarks, Materials, Assignments 
8/25  Introduction. Preliminaries. (Ch. 1) 
Introduction notes Sets and Equivalence Relations 
9/1  No class on 9/2 (Labor Day) Integers. (Ch. 2) 
Induction and WellOrdering Principle Division Algorith, Euclidean Algorithm, Prime Numbers Homework 1 due 9/6: All students: Ch. 1, §1.3 "Exercises" (online text), exer. 7, 9, 18(b,c), 19, 20(b), 21, 22(c,e), 25(d). Graduate (MTH 431) students: in addition, do exer. 26 and 29. (MTH 331 students are welcome to do these problems for extra credit.) Latex template for homework: here. Homework 1 Solutions 
9/8  Groups. (Ch. 3)  Integer Equivalence Classes and Symmetries Homework 2 due 9/13: All students: Ch. 2, §2.3 "Exercises" (online text), exer. 2, 5, 12, 15(d, f), 18. Graduate (MTH 431) students: in addition, do exer. 17(c) and 23. (MTH 331 students are welcome to do these problems for extra credit.) Latex template for homework: here. Homework 2 Solutions 
9/15  Groups Cont'd. Subgroups. (Ch. 3) 
Groups: Definitions and Examples Summary of Some Groups Examples Subgroups Homework 3 due 9/20: All students: Ch.2, §2.3, exer. 28, 31; Ch. 3, §3.4, exer. 1 (b,d), 2 (c,d), 6, 7, 8, 14. Graduate (MTH 431) students: in addition, do exer. 10 and 15 in Section 3.4. (MTH 331 students are welcome to do these problems for extra credit.) Latex template for homework: here. Homework 3 Solutions 
9/22  Exam I on 9/23 (Monday) Exam I solutions: MTH 331 version MTH 431 version Cyclic Groups. (Ch. 4) 
Cyclic Groups Study for Exam I (9/23): study guide Homework 4 due 9/27: All students: Ch. 3, §3.4, exer. 16, 26, 33, 39, 43, 48. Graduate (MTH 431) students: in addition, do exer. 34 and 47. (MTH 331 students are welcome to do these problems for extra credit.) NOTE: Extra problem for all: 45. Latex template for homework: here. Homework 4 Solutions Sage Lab on Thursday: explore "Sage" Sections 1.5, 2.6, and 3.7. 
9/29  Cyclic Groups Cont'd. Permutation Groups. (Ch. 5) 
Subroups of the Circle Group Homework 5 due 10/7 (Monday): All students: Ch. 4, §4.4, exer. 1 (a,b,c), 2 (a,b,d), 3 (b,c,h,k), 4 (d), 11, 23 (a), 31. Graduate (MTH 431) students: in addition, do exer. 2 (f), 23 (c), and 26. (MTH 331 students are welcome to do these problems for extra credit.) NOTE: Extra problem for all: 32. Latex template for homework: here. Homework 5 Solutions 
10/6  Permutation Groups Cont'd. Fall Recess 10/10 and 10/11: no classes! 
Permutation Groups Dihedral Groups Homework 6 due 10/16: All students: Ch. 4, §4.4, exer. 6; Ch. 5, §5.3, exer.1, 2 (b,d,g,h,j,p), 3 (b,d), 4, 22, 23, 24, 27. Graduate (MTH 431) students: in addition, do exer. 13 in Section 5.3. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 6 and 34 in Section 5.3. Latex template for homework: here. Homework 6 Solutions 
10/13  Exam II on 10/17 (Thursday) Exam II solutions: MTH 331 version MTH 431 version Cosets and Lagrange's Theorem (Ch. 6) 
Study for Exam II (10/17): study guide
Homework 7 due 10/25: All students: §6.4, exer. 1, 3, 4, 5 (a,b,c), 8, 11 (only prove statements (a) and (d) are equivalent), 17. Graduate (MTH 431) students: in addition, do exer. 18, 19. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 5 (f), 13. Latex template for homework: here. Homework 7 Solutions 
10/20  Cosets and Lagrange's Theorem Cont'd. Fermat's and Euler's Theorems. Isomorphisms (Ch.9) 
Cosets and Lagrange's Theorem Fermat's and Euler's Theorems 
10/27  Isomorphisms Cont'd. Direct Products (Ch.9) 
Isomorphisms: Definitions and Examples Direct Products Homework 8 due 11/6: All students: §9.3, exer. 2, 3, 7, 8, 9 (do not need to proof that G is a group, prove isomorphism only), 16 (b and d), 34, 46, 50. Graduate (MTH 431) students: in addition, do exer. 24 and 41. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 15, 25. Latex template for homework: here. Homework 8 Solutions 
11/3  Normal Subgroups and Factor Groups. (Ch.10) Homomorphisms (Ch.11) 
Normal Subgroups. Factor Groups. Simple Groups. Homework 9 is due 11/13: All students: §10.3, exer. 1 (a,b,c,e), 5, 6, 8, 9, 13 (a,c). Graduate (MTH 431) students: in addition, do exer. 11 and 12. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 13 (b,d). 
11/10  Homomorphisms Cont'd.  Study for Exam III (11/14): study guide 
11/17  
11/24  Tuesday follows Thursday schedule Thanksgiving Recess: no classes 11/2729 

12/1  Student presentations: list of topics  
12/8 
Monday follows Friday schedule Final Exams start 12/10/19 
