Math 301 (Spring 2024)

• Exam 3 is on Wednesday, May 22 1:45–3:45pm in Kiely 320. The exam will cover Week 10 through Week 15, which includes HW 9, 10, 11, 12. The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exam that includes only definitions and theorem/proposition statements from class and the textbook. 

• Solutions to Exam 3

• Suggested reading: Section 11.2, Section 16.1, Section 16.2

• Homework Assignment #12. This assignment will not be collected, but the problems are fair game for Exam 3. 

• Monday's class: Proved the first isomorphism theorem and Cauchy's theorem for finite abelian groups.

• Wednesday's class: Introduced rings and there various properties rings can have, which leads to the defition of a field. 

Suggested reading: Section 9.2, Section 10.1 

Required Reading: Notes on normal subgroups and homomorphisms (TeX).  These notes are a replacement for our missed class meeting on Wednesday. 

• We will not have class on Wednesday, April 17.  However, I encourage you use the time and classroom to meet your peers and discussed the required reading.  

• There will be no office hours this week on account of my travels.  Please post questions to Discord.  

• Homework Assignment #10. Due Wednesday, May 1.  

• Solutions to Exam 2

• Monday's class: Introduced the notion of the internal direct product and discussed examples. Discussed, but did not prove, the fundamental theorem of finite abelian groups. 

Suggested reading: Section 9.1, Section 9.2

• Office hours Wednesday moved to 3:30–4:30pm.  If this affects your ability to complete your homework, you may drop it in my mailbox or under my office door by end of day Friday.

• Exam 2 is on Wednesday, April 10.  The exam will cover the course material covered Week 5 through Week 9, which includes HW5, HW6, HW7, and HW8.  The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exam that includes only definitions and theorem/proposition statements from class and the textbook (up through Section 6.3).  

• Homework Assignment #9. Due Wednesday, April 17.  

Monday's class:  Introduced the notion of an isomorpism, went over examples, discussed basic properties, and proved that every infinite cyclic group is isomorphic to the integers. 

Wednesday's class: Proved the classification of cyclic groups.  Proved Cayley's theorem.  Proved that all finite groups are linear. 

Suggested reading: Section 5.1, Section 5.2

Office hours Wednesday moved to 3:30–4:30pm.  If this affects your ability to complete your homework, you may drop it in my mailbox or under my office door by end of day Friday.

• Homework Assignment #7. Due Wednesday, March 27. 

Monday's class: Gave a formula for the order of a permutation.  Defined the notion of an even/odd permutation and showed that the notion is well defined.  Defined the alternating group. 

Wednesday's class: Computed the order of the alternating group A_n.  Introduced the notion of a graph and graph automorphism.  Defined the dihedral group D_n as a group of graph automorphism.  Computed the order of D_n.  Proved that D_n can be generated by two elements.  

Suggested reading: Section 3.3, Section 4.1

• Exam 1 is on Wednesday, March 6.  The exam will cover the course material through Week 4, which includes HW1, HW2, HW3, and HW4.  The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exams that includes only definitions and theorem/proposition statements from class and the textbook (up through Section 3.2).  

• Homework Assignment #5. Due Wednesday, March 13. 

Monday's class: Continued out treatment of cyclic groups, proving that every subgroup of a cyclic group is cyclic. Introduced the notion of an order of an element of a group.  Computed the order of elements in cyclic groups. 

Suggested reading: Section 3.2

• Note:  We have class on Thursday, February 22, as CUNY will follow a Monday Schedule. 

Homework Assignment #4. Due Wednesday, Februrary 28. 

Monday's class: Introduced the notion of a group and discussed basic examples.

Wednesday's class: Introduced a more extensive list of exampls of groups, and we established some basic properties of groups.  Our class was interrupted by a campus emergency, and so please make sure read Section 3.2.  In particular, we were in the process of finishing the proof of Proposition 3.21 when class was interrupted.  I was going to finish the lecture portion of the class with stating an immediate corollary of Prop 3.21, which is Prop. 3.22 in the book. 

Suggested reading: Section 2.1 and Section 2.2 

Homework Assignment #2. Due Wednesday, Februrary 14. 

Monday's class: Introduced the greatest common divisor (gcd) of two integers.  Proved that the gcd is a linear combination.  Established the Euclidean algorithm for finding the gcd and discussed how to write the gcd as a linear combination using the steps from the Euclidean algorithm. 

Wednesday's class: Introduced prime numbers.  We proved Euclid's lemma, we proved that every natural number other than one is a product of prime numbers (and stated the Fundamental Theorem of Arithmetic), and we proved there are infinitely many primes.  We wrote down the principal of mathematical induction (which was proved on HW1).