Important Dates

MATH 3140-001: Abstract Algebra 1

Spring 2021

MWF 9:10-10:00 am, REMOTE (Zoom)

Schedule

Not all material covered in class can be found in the textbook.

Go to the schedule for the current week.

WEEK 1

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Jan 15		Syllabus.
Jan 18		Martin Luther King Jr. Holiday. No class.
Jan 20	Sections 1.1-1.3 of the text.	Symmetries of a rectangle and a square; multiplication tables of their symmetries.	Lecture notes on Canvas; Sections 1.1-1.3 (pp. 1 - 11).
Jan 22	Section 1.4 and Section 1.5 (first half: pp. 16 - mid 19)	Symmetries and matrices; symmetries and permutations.	Lecture notes on Canvas; Sections 1.4 -- 1.5 (pp. 11 - mid 19).

WEEK 2

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Jan 25	Sets, Relations, and Functions (Review)	Axioms for sets. Relations and functions.	Background on Sets, Relations, and Functions Lecture notes on Canvas. Quiz 1 (on Canvas, due: 9 am on Jan 27)
Jan 27	Review methods of proof (direct proof, proof of the contrapositive, proof by contradiction).	Constructing and communicating proofs.	Lecture notes on Canvas.
Jan 29	Review proof by induction and by strong induction.	Natural numbers and induction.	Background on the Natural Numbers and Induction Lecture notes on Canvas. Homework 1 (on Canvas, due: 9 am on Feb 8) Graded in DS part: Pr2 Information about Homework Assignments

WEEK 3

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Feb 1		Group work: Worksheet 1: Sets, Relations, Functions, and the Natural Numbers	Work on the problems that were not solved in class. Solution key to Worksheet 1 on Canvas. Quiz 2 (on Canvas, due: 9 am on Feb 3)
Feb 3	Review what you learned in earlier courses about finite sets and infinite sets.	Finite sets, infinite sets.	Background on Finite Sets and Infinite Sets Lecture notes on Canvas.
Feb 5	Section 1.5 (pp. mid 19 - 22).	Permutations of finite sets.	Lecture notes on Canvas. Section 1.5 (pp. mid 19 - 22). Homework 2 (on Canvas, due: 9 am on Feb 15) Graded in DS part: Pr2 Information about Homework Assignments

WEEK 4

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Feb 8	Section 1.6 (pp. 24 - 25).	The integers: constructing $\mathbb{Z}$ from $\mathbb{N}$.	Lecture notes on Canvas. Section 1.6 (pp. 24 - 25). Quiz 3 (on Canvas, due: 9 am on Feb 10)
Feb 10	Section 1.6 (pp. 26 - mid 29).	The integers: divisibility and prime factorization.	Lecture notes on Canvas. Section 1.6 (pp. 26 - mid 29).
Feb 12	Section 1.6 (pp. mid 29 - 33).	The integers: the Euclidean algorithm and the uniqueness of prime factorization.	Lecture notes on Canvas. Section 1.6 (pp. mid 29 - 33). Homework 3 (on Canvas, due: 9 am on Feb 22) Graded in DS part: Pr2(b), Pr3(a) Information about Homework Assignments

WEEK 5

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Feb 15	Section 1.7.	Modular arithmetic.	Lecture notes on Canvas. Section 1.7 (pp. 37 - mid 42). Quiz 4 (on Canvas, due: 9 am on Feb 19)
Feb 17		Wellness Day. No class.
Feb 19	Section 1.7 (pp. mid 42 - 43), Section 1.9 (Prop. 1.9.9), and Sections 1.10 - 1.11	Groups, rings, and fields	Lecture notes on Canvas. Section 1.7 (pp. mid 42 - 43), Section 1.9 (Prop. 1.9.9), and Sections 1.10 - 1.11 Homework 4 (on Canvas, due: 9 am on March 1) Graded in DS part: Pr2(a),(b) Information about Homework Assignments

WEEK 6

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Feb 22	Section 2.1.	Groups: first consequences of the definition. The general associative law. Groups of small order.	Lecture notes on Canvas. Section 2.1. Quiz 5 (on Canvas, due: 9 am on Feb 24)
Feb 24	Section 2.2 (pp. 94 - mid 99).	Subgroups and their generating sets. Cyclic groups.	Lecture notes on Canvas. Section 2.2 (pp. 94 - top 98).
Feb 26	Section 2.2 (top 98 - mid 103).	The order of a group element. Classification of cyclic groups up to isomorphism. Subgroups of cyclic groups.	Lecture notes on Canvas. Section 2.2 (top 98 - mid 103). Practice Problems on Groups. No Homework

WEEK 7

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Mar 1		Completing the proof of the theorem on subgroups of finite cyclic groups. Group work: Worksheet 2: Groups, Subgroups, Cyclic Groups.	Lecture notes on Canvas. Solution key to Worksheet 2 on Canvas. Quiz 6 (on Canvas, due: 9 am on Mar 3).
Mar 3	Review for the Midterm.	Review for the Midterm.	Answer key to the practice problems on the review sheet.
Mar 5		Midterm	No Homework

WEEK 8

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Mar 8	Section 2.3.	The orthogonal group $O(2)\cong O(2,\mathbb{R})$ and the dihedral groups $D_n$.	Lecture notes on Canvas. Section 2.3 No Quiz.
Mar 10	Section 2.4 (pp. 111 - mid 115, mid 117 - top 118)	Homomorphisms: Examples and first consequences of the definition.	Lecture notes on Canvas. Section 2.4 (pp. 111 - mid 115, mid 117 - top 118) Note: The proof of Prop. 2.4.11(a) in the text is incorrect.
Mar 12	Section 2.4 (pp. mid 115 - mid 117) Review equivalence relations and partitions: see Lec.Notes 1/25 and Section 2.6 (Def. 2.6.1 and pp. 128 - 131).	Further properties and examples of homomorphisms. The kernel of a homomorphism.	Lecture notes on Canvas. Section 2.4 (pp. mid 115 - mid 117). [Review of equivalence relations and partitions: Lec.Notes 1/25 and Section 2.6 (Def. 2.6.1 and pp. 128 - 131).] Homework 5 (on Canvas, due: 9 am on March 29) Graded in DS part: Pr2(b), Pr3(b) Information about Homework Assignments

WEEK 9

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Mar 15		No class. Campus is closed, due to winter storm impacts.
Mar 17	Section 2.5, Section 2.6 (p. 133)	Normal subgroups, conjugation, and conjugacy classes. Lagrange's Theorem.	Lecture notes on Canvas. Section 2.5, Section 2.6 (p. 133) Quiz 7 (on Canvas, due: 9 am on Mar 19).
Mar 19		Group work: Worksheet 3: Normal Subgroups and Conjugation. Lagrange's Theorem	Work on the problems that were not solved in class. Solution key to Worksheet 3 on Canvas.

WEEK 10 (Spring Pause)

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Mar 22	Section 2.7 (pp. 134 - mid 140, skip Examples 2.7.11-12).	Quotient groups and the homomorphism theorem.	Lecture notes on Canvas. Section 2.7 (pp. 134 - mid 140, skip Examples 2.7.11-12). No Quiz.
Mar 24	Section 2.7 (pp. 142 - top 146, skip Example 2.7.21).	The correspondence theorem and isomorphism theorems involving quotient groups.	Lecture notes on Canvas. Section 2.7 (pp. 142 - top 146, skip Example 2.7.21).
Mar 26		Group work: Worksheet 4: Quotient Groups and the Homomorphism/Isomorphism Theorems	Work on the problems that were not solved in class. Solution key to Worksheet 4 on Canvas. Homework 6 (on Canvas, due: 9 am on Apr 5) Graded in DS part: Pr2 Information about Homework Assignments

WEEK 11

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Mar 29	Section 3.1 (pp. 149 - mid 152).	Direct products of two groups. The Chinese Remainder Theorem.	Lecture notes on Canvas. Section 3.1 (pp. 149 - mid 152). Review Prop. 1.7.9, 1.11.7, and Ex. 2.7.12. Quiz 8 (on Canvas, due: 9 am on Mar 31).
Mar 31	Section 3.1 (pp. mid 152 - 154) and Section 3.2 (pp. 160 - mid 162).	Direct products of $k$ groups. Semidirect product.	Lecture notes on Canvas. Section 3.1 (pp. mid 152 - 154) and Section 3.2 (pp. 160 - mid 162).
Apr 2		Finite abelian groups.	Lecture notes on Canvas. [Optional: Section 3.6 (3.6.7 - 3.6.20, 3.6.21 [the name is correctly: Fundamental Theorem of Finite Abelian Groups], 3.6.24)] Homework 7 (on Canvas, due: 9 am on Apr 12) Graded in DS part: Pr2 Information about Homework Assignments

WEEK 12

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Apr 5		Group work: Worksheet 5: Direct Product and Finite Abelian Groups	Work on the problems that were not solved in class. Solution key to Worksheet 5 on Canvas. Quiz 9 (on Canvas, due: 9 am on Apr 7).
Apr 7	Section 5.1 (pp. 242 - mid 246).	Group actions.	Lecture notes on Canvas. Section 5.1 (pp. 242 - 244).
Apr 9	Section 5.1 (pp. 245 - mid 246), Section 5.3.	Group actions and symmetries of groups.	Lecture notes on Canvas. Section 5.1 (pp. 245 - mid 246), Section 5.3. Homework 8 (on Canvas, due: 9 am on Apr 19, last homework) Information about Homework Assignments

WEEK 13

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Apr 12	Section 5.4 (pp. 255 - 259).	Applications of the Class Equation: Sylow's 1st Theorem and finite $p$-groups.	Lecture notes on Canvas. Section 5.4 (pp. 255 - 259). Quiz 10 (on Canvas, due: 9 am on Apr 14).
Apr 14	Section 5.4 (pp. mid 259 - 263).	Applications of Sylow's theorems. Groups of small order.	Lecture notes on Canvas. Section 5.4 (pp. mid 259 - 263).
Apr 16		Group work: Worksheet 6: Groups of small order: Orders 8 and 12	Work on the problems that were not solved in class. Solution key to Worksheet 6 on Canvas.

WEEK 14

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Apr 19	Section 1.8.	The ring $K[x]$ of polynomials over a field $K$.	Lecture notes on Canvas. Section 1.8 (pp. 45 - mid 51). Quiz 11 (on Canvas, due: 9 am on Apr 21; last quiz).
Apr 21	Section 1.8 (pp. mid 51 - 54)	Unique factorization in $K[x]$ for any field $K$. Evaluation homomorphisms $K[x]\to K$ and roots of polynomials.	Lecture notes on Canvas. Section 1.8 (pp. mid 51 - 54), Section 6.2 (pp. bottom 276 - top 278, except Prop. 6.2.6).
Apr 23		Finite subgroups of $K^*$ for a field $K$. Ring homomorphisms and quotient rings.	Lecture notes on Canvas. Section 3.6 (Thm. 3.6.25), Section 6.2 (pp. 275 - bottom 276, mid 280 - mid 281), Section 6.3 (pp. bottom 288 - bottom 290, except Example 6.3.3)

WEEK 15

Date	Study before class	Topics covered in class	Notes/reading and assignments after class
Apr 26		Ideals and quotient rings of $K[x]$ for any field $K$.	Lecture notes on Canvas. Section 6.2 (Prop. 6.2.29), Section 6.3 (Examples 6.3.3 and 6.3.5).
Apr 28	Review for the Final Exam.	Review for the Final Exam.	Answer key to the practice problems on the review sheet.
Apr 30		Reading Day. No class.