### MAT 511 Abstract Algebra I

(last update 12/9, 3:00 pm)

The final exam takes place Monday December 11, in Room 207 of Adel Mathematics. It is comprehensive, covering all material discussed in class or appearing on exams, problem sets, or in assigned exercises from the text, and including material on supplementary handouts.

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I made an annotated list of exam and homework problems that deserve special attention in your preparations for the final exam - sorry for the rudimentary formatting:

Relevant material for final exam study:

Exams

Exam 1: 1a,c,e (see also HW 3.3a and 3.4), 2, 3 and 4 (all related), 5 (anticipating semi-direct products)

Exam 2: 1(a), 2, 3a, relating to the ``semi-dichotomy” between simple groups and solvable groups, e.g., for the family of groups of order \$p^aq^b\$ (see also HW 4.2d), 3b, generalizing the family of p-groups to nilpotent groups - note 2a and 2b extend easily to the family of nilpotent groups, 4 (see also HW 7.5), 5, 6a, and 6b, as corrected - the tools used in the solutions are useful in many situations (6b is used to handle one case in the proof that groups of order p2q2 are not simple)

Exam 3: 1 (esp. with regard to the the definition of sign of a permutation and the alternating group), 2, 3a (see Chapter 4 of the text for more on permutation characters), 3b (see Isaacs’ video (Lecture 3 I believe) for the computation of the regular character), 3c - it’s important to understand the notion of reducible and irreducible representation (and character) - this is a simple example, in fact, of a completely reducible representation, 4, 5a, b

Problem Sets

- HW 1: 4 (note the connection between part b and the last steps in the proof sketch of the Wedderburn-Artin Theorem)
- HW 2: 1 , 2, 3 (the N/C theorem), 4a, b (this is a wonderful example of a semi direct product), 5a (the equivalence relation defining the partition of a group by cosets of a subgroup), 5b (the second ingredient in the proof of Lagrange’s Theorem), 6 (the first isomorphism theorem for groups - see also HW 3.3a)
- HW 3: 1 (only in the special case the K is normal, so HK is a subgroup of G, in which case the formula in part d follows from the second isomorphism theorem), 2, 3a (a very useful result), 3b (the defining property of the commutator subgroup and abelianization of a group), 4 (see also Exam 1.1e)
- HW 4: 1 (the third isomorphism theorem), 2 (metabelian and solvable groups), 3a plus HW 5.3 (Cauchy’s Theorem), 4 (in connection with semi-direct products)
- HW 5: 1 (only in connection with Sylow subgroups, which are pi-subgroups where pi consists of a single prime), 3 (see HW 4.3a), 4 (see also Exam 2.4 and HW 7.5), 5
- HW 6: 1 and 2 (esp. in connection with HW 3.3(a), see also Exercise G), 3 (first isomorphism theorem for modules), 4 (esp. in connection with the decomposition of the group ring of a finite group)
- HW 7: 1 (in connection with Jacobsen radical and semi simple rings), 2 (only for the definition of ideal and method of establishing ideal-ness), 3c (the kernel of a ring homomorphism is an ideal), 4, 5 (see also Exam 2.4 and HW 5.4
- HW 8: 1 (with the idea of doing the same kind of thing for other groups or rings), 2 (a good review of group-theoretic tools)
- Exercises: A, D, E, F (in connection with semi direct products of groups, and complete reducibility of modules), H (in for the proof of Cauchy’s Theorem using the Sylow E Theorem); the most important exercises from the text appeared on problem sets

Exam 3 takes place in class on Friday, 12/1. There is a review session Thursday, 11/30, in Room 162, from 4:30 until all questions are answered. The material to be covered on the exam is the material on rings and modules has been covered in class and on problem sets and exercises, since Exam 2 was distributed (October 28), and the material covered in Lectures 1 and 2 of Isaacs' videos (just up to the Exercises 2.1 and 2.2). (I hope you have studied further than that, but there won't be anything on the exam from beyond that point.) This includes most of Chapter 12 of Isaacs' book - anywhere you see the phrase "abelian X-group" replace it with "right R-module." (Also he writes a plus sign with a dot above it to denote direct sum.) Also I've uploaded Thomas Holtzworth's M.S. thesis, linked below - you will find that much of what we have done is treated in Chapter 1.

Also here is a pdf of Isaacs' Character Theory book. If you find it interesting reading, I encourage you to purchase an inexpensive copy: click here.

Isaacs' Character Theory

Other resources include these books:

• Character Theory, by L. Grove
• Noncommutative Rings, by I.N. Herstein
• any other graduate algebra text, such as Hungerford, Algebra

I've written up some supplementary notes to accompany Isaacs' video lectures - they include review material on rings from undergraduate abstract algebra, along with other background material to make the lectures more accessible, some of which has been discussed in class and some which will be over the coming lectures. I will re-post the notes as they develop - here is the first installment. I will add to these notes over the weekend.

Supplemental notes on rings, modules, and representations

Some comic relief, possibly amusing (produced by some grad students with too much time on their hands):

Finite simple group of order two

Here is a link to the first of the video lectures of Isaacs on representation theory and group characters:

Isaacs' Lecture 1

Isaacs' Lectures

Our goal is to get through the proof of Burnside's paqb theorem, at the end of Chapter 5, which occurs in the third 75-minute lecture. We will attempt to get through one lecture per week, so you can start by watching the first part of Lecture 1. Here is a pdf of the slides, trimmed thanks to Jeffrey C, including only Chapters 1-5, which is what we hope to finish by December 8:

Isaacs' slides (trimmed)

Solutions to HW #5 are now available on the BBLearn page.

Here is a pdf with statements of Exercises assigned in class (including Exercises A-F):

Exercises

Isaacs Chapter 1

Isaacs Chapter 2

Problem Sets and Solutions - see BBLearn page

Exercises

Ch. 1 / 1.1, 1.5, 1.7. (Solutions: 1.1)

The exercises listed above from Chapter 1 are due Monday, 9/18.

Ch. 2 / 2.3, 2.4, 2.5, 2.6.

Solutions: 2.3, 2.4, 2.5 , 2.6

These exercises listed above from Chapter 2 are due Wednesday, 9/20.

Ch. 2 / 2.8, 2.9, 2.11, 2.12. (2.8 was included on HW #3.)

Ch. 2 / 2.13, 2.16, 2.18, 2.20, 2.21. ( 2.13(b) and 2.20 were done in class, and 2.13(c) is equivalent to Exam 1.5(c).); 2.19 and 2.22 were included, in modified form, on HW #2, 2.16 was on HW #3.)

Solutions: 2.18, 2.21

The exercises listed above from Chapter 2, along with in-class exercises A and B, are due Friday, 10/13.

Ch. 3 / 3.1, 3.3. (Both these problems were on HW #3.)

Ch. 3 / 3.2, 3.4, 3.5, 3.9, 3.11, 3.13, 3.14 (3.9 was done in class, as a corollary of HW #2.3)

The exercises listed above from Chapter 3 (except 3.1, 3.3, and 3.9), along with in-class exercises C and D, are due Wednesday, 10/25.

Ch. 4 / 4.1