(Part 2) Best abstract algebra books according to redditors

I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.

On the CS side:

• Basic automata/formal languages/Turing machines; Sipser is recommended here.
  
• Basic programming language theory; I used University of Washington CSE P505 online video lectures and materials and can recommend it.
  
• Formal semantics; Semantics with Applications is good.
  
• Compilers. You'll need several resources for this; my personal favorites for an introductory text are Appel's ML book or Programming Language Pragmatics, and Muchnick is mandatory for an advanced understanding. All of the graph theory that you need for this type of work should be covered in books such as these.
  
• Algorithms. I used several books; for a beginner's treatment I recommend Dasgupta, Papadimitriou, and Vazirani; for an intermediate treatment I recommend MIT's 6.046J on Open CourseWare; for an advanced treatment, I liked Algorithmics for Hard Problems.
  
  On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
  
  
• You run into abstract algebra a lot in program analysis as well as in cryptography, so it's best to begin with a solid foundation along those lines. There's a lot of debate as to what the best text is. If you're never touched the subject before, Gallian is very approachable, if not as deep and rigorous as something like Dummit and Foote.
  
• Order theory is everywhere in program analysis. Introduction to Lattices and Order is the standard (read at least the first two chapters; the more you read, the better), but I recently picked up Lattices and Ordered Algebraic Structures and am enjoying it.
  
• Complexity theory. Arora and Barak is recommended.
  
• Formal logic is also everywhere. For this, I recommend the first few chapters in The Calculus of Computation (this is an excellent book; read the whole thing).
  
• Computability, undecidability, etc. Not entirely separate from previous entries, but read something that treats e.g. Goedel's theorems, for instance The Undecidable.
  
• Decision procedures. Read Decision Procedures.
  
• Program analysis, the "accessible" variety. Read the BitBlaze publications starting from the beginning, followed by the BAP publications. Start with these two: TaintCheck and All You Ever Wanted to Know About Dynamic Taint Analysis and Forward Symbolic Execution. (BitBlaze and BAP are available in source code form, too -- in OCaml though, so you'll want to learn that as well.) David Brumley's Ph.D. thesis is an excellent read, as is David Molnar's and Sean Heelan's. This paper is a nice introduction to software model checking. After that, look through the archives of the RE reddit for papers on the "more applied" side of things.
  
• Program analysis, the "serious" variety. Principles of Program Analysis is an excellent book, but you'll find it very difficult even if you understand all of the above. Similarly, Cousot's MIT lecture course is great but largely unapproachable to the beginner. I highly recommend Value-Range Analysis of C Programs, which is a rare and thorough glimpse into the development of an extremely sophisticated static analyzer. Although this book is heavily mathematical, it's substantially less insane than Principles of Program Analysis. I also found Gogul Balakrishnan's Ph.D. thesis, Johannes Kinder's Ph.D. thesis, Mila Dalla Preda's Ph.D. thesis, Antoine Mine's Ph.D. thesis, and Davidson Rodrigo Boccardo's Ph.D. thesis useful.
  
• If you've gotten to this point, you'll probably begin to develop a very selective taste for program analysis literature: in particular, if it does not have a lot of mathematics (actual math, not just simple concepts formalized), you might decide that it is unlikely to contain a lasting and valuable contribution. At this point, read papers from CAV, SAS, and VMCAI. Some of my favorite researchers are the Z3 team, Mila Dalla Preda, Joerg Brauer, Andy King, Axel Simon, Roberto Giacobazzi, and Patrick Cousot. Although I've tried to lay out a reasonable course of study hereinbefore regarding the mathematics you need to understand this kind of material, around this point in the course you'll find that the creature we're dealing with here is an octopus whose tentacles spread in every direction. In particular, you can expect to encounter topology, category theory, tropical geometry, numerical mathematics, and many other disciplines. Program analysis is multi-disciplinary and has a hard time keeping itself shoehorned in one or two corners of mathematics.
  
• After several years of wading through program analysis, you start to understand that there must be some connection between theorem-prover based methods and abstract interpretation, since after all, they both can be applied statically and can potentially produce similar information. But what is the connection? Recent publications by Vijay D'Silva et al (1, 2, 3, 4, 5) and a few others (1 2 3 4) have begun to plough this territory.
  
• I'm not an expert at cryptography, so my advice is basically worthless on the subject. However, I've been enjoying the Stanford online cryptography class, and I liked Understanding Cryptography too. Handbook of Applied Cryptography is often recommended by people who are smarter than I am, and I recently picked up Introduction to Modern Cryptography but haven't yet read it.
  
  Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.

There's this great math textbook that I used called "Introduction to Analysis and Abstract Algebra" by a guy named Hafstrom that I found in the basement of my university library.

It was really cool because it treated the two subjects simultaneously, each supporting the other, whereas normally they're kept separate. It's been a few years since I've read it but as I recall the theoretical development starts with explaining what a field is (groups and rings come later) and explaining how the real numbers form a field with just a few extra axioms. It's a very different approach than what I encountered in my analysis and abstract algebra courses.

https://www.amazon.com/Introduction-Analysis-Abstract-Algebra-Hafstrom/dp/0721644554

If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.

Here's a brief outline plus some resources:

First thing to know is that since simple groups have no normal subgroups, they cannot be the kernel of any non-trivial have a non-trivial kernel for any morphism (since the kernel of a morphism is normal). This means that any morphism from a simple group G, is either 1-1 or it's the trivial map. As a consequence, simple groups are in some sense "atomic", you can't map them to a smaller group unless you completely crush them. As such they serve as the "building blocks" of all other groups. If you could classify all such groups, you would have a list of all the "atoms" of finite group theory.

Around 1910, Burnside conjectured that all non-abelian finite simple groups must have even order (I don't have a reference for this). This was eventually proved by Feit-Thompson) in 1962. Around 1970, Danny Gorenstein announced a program for the classification of all non-abelian finite simple groups. In 1983, he announced that the proof, carried out by hundreds of mathematicians and spanning many thousands of pages, was complete. Then, being the only one with the full picture of the proof, he promptly died.

Here in an intro written for physicists, but very useful.

For a more serious exposition (i.e. grad school level class) you might see Wilson's The Finite Simple Groups.

To get a feel for the classification proof and the way you might apply the classification see Stephen Smith's talk

For just the constructions of the groups you might look here. It's worth noting that when giving a "construction", one is looking for a sort of "explanation" of the group (other than giving a list of permutations), i.e. Group G occurs as the automorphism group of object X.


EDIT: For the Monster, the "natural" object on which it acts is a vertex algebra and you can see the construction of the one in question in "Vertex operator algebras and the Monster" by I. Frenkel, Lepowsky, and Meurman

If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):

Basic Algebra by Jacobson

Algebra by Lang

Algebra by MacLane/Birkhoff

Algebra by Herstein

Algebra by Artin

etc

But there are other books that are "essential" to modern readers:

Chapter 0 by Aluffi

Basic Algebra by Knapp

Algebra by Dummit/Foot

Basically, another way to define them is any subset of N of the form S = {t1a1 + t2a2 + ... + tnan | t1, t2, ..., tn are in N} and gcd(a1, a2, ..., an) = 1, where a1, ..., an are fixed. This set contains 0 and is closed under addition, so it's a submonoid of N, and you can prove that the complement is always finite, which is where the gcd assumption is used. This definition also shows you the connection to linear Diophantine equations with nonnegative coefficients.

One fun application is to something called the Coin problem, which asks for the largest amount of money you can't get by adding up given types of coins. There's also a special case called the McNugget numbers. You might be able to guess where that one comes from haha.

There's also all kinds of connections to algebraic geometry and rings with non-unique factorization (for example, in the semigroup generated by <2,3>, 6 = 2 + 2 + 2 and 3 + 3, so you can't "factor" things uniquely), but I don't know much more than that.

If you want to get really in depth, there's this book which I'm really enjoying so far.

I'm not a condensed matter guy, so I don't have a resource that covers that. But the best low graduate level book on group theory that I have found is by Matthew Robinson, Symmetry and the Standard Model. This book was where group theory really clicked for me. This guy is a fantastic author.

Hungerford has both an undergraduate text and his GTM text. The undergraduate text starts off with ring theory and then moves to group theory. The GTM text starts with group theory and moves onto ring/field theory.

Yeah I linked the Algorthmics Lab more for what they are doing, not so much the general courses. I lost track of what became of Wolfram's ideas, so it's great to find the resource. So entrenched in Kolmogorov complexity theory though ....... ugh. I find Kolmogorov particularly challenging.

I think both top-down and bottom-up are essential. Understanding the landscape of fields goes hand-in-hand with hammering through fundamentals. Even just quick bibliography -> abstract -> conclusion skimming through overview paper or papers from citation lists flowing off seminal papers will do wonders to help put fundamentals in context. Google Scholar is your friend. lol. But I guess getting 200-300 level math nice and solid really is the first step.

The Witness!! Great game. It's interesting, I was thinking about building mental models the other day, how Feynman talked about it, and how I should be cultivating that more. I think that's why interdisciplinary work is so interesting, so much can be discovered there (the alchemy plays out differently, la), and it's due to the mixing of different mental models.

Choosing one book is tough. The foundations are analysis and algebra, and it was my introduction to those topics that grabbed me. This brilliant self-contained analysis book starts with the algebraic axioms of the real numbers, and ends with Lebesgue integration. (The other analysis book at this level would be baby Rudin). This book goes from nothing to everything you need to know about series of functions, inner product spaces, and fourier series to tackle pretty much any higher level textbook.

https://www.amazon.ca/Foundations-Mathematical-Analysis-Richard-Johnsonbaugh/dp/0486477665/ref=sr_1_1?keywords=pfaffenberger&qid=1558986484&s=books&sr=1-1

As for algebra, I had the luck of being taught by the author of this concise textbook (Dummit & Foote would be the other one to look at):

https://www.amazon.ca/Abstract-Algebra-Introduction-Groups-Applications/dp/9814730548/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1558986393&sr=8-1

There seems to be generally agreed upon seminal texts for every field of study. Knuth for compsci, Rudin for analysis, Hatcher for algebraic topology, Dummit & Foote for abstract algebra, Feller for probability, Bollobas for modern graph theory, etc etc.

Out of curiosity, may I ask the books you have lined out? Once you start to choose a specific topic of study, the options tend to become exponential ...

I like Benjamin Steinberg's Representation Theory of Finite Groups. It is small and at an advanced undergraduate level.

> only a 4 or 5 people in her set 1 class even understand algebra.

May I then recommend A Survey of Modern Algebra by Garrett Birkhoff and Saunders Mac Lane.

If you like the online course lectures, you should definately look at those. I know tons of great schools such as Yale, UCLA, MIT, Stanford etc. etc. offer full lecture series on youtube. Usually the syllabi are online for you to look at so you can get a feel for it.

I am more of a book learner myself so I will try to make some recommends, but when looking for books try googling, reading stackexchange posts and Amazon reviews.

I'm going to disagree with /u/Orion952 on Fraleigh's book, its an alright book but I have seen much better. For Abstract Algebra, I would recommend Nicholson's book. Its a very gentle introduction to the subject. There are lots of computation problems as well as proofs you can work through so you can get a nice feel for the subject. I would also hunt down the pdf for Dummit and Foote's book as well, I thought it was pretty gentle for the most part as well as comprehensive.

For analysis and topology, I have encountered some decent books.

Strichartz for analysis is very wordy and conversational, so I didn't care for it myself hence didn't read very much of it (I much prefer the style of Walter Rudin) but it might be good for starting out.

Bhatt has written a very nice book for analysis and covers a lot of material on metric space topology. I actually know the author pretty well so if you are interested in the book I may be able to hook you up.

Simmons has written a book that has a pretty conversational style, but I wasn't a big fan of his style. Bhatt's book will have a more "traditional" approach, but thats not to say it isn't readable. The first half of the book will cover the same stuff Bhatt's book does and the second half will be more advanced stuff including some concepts from Functional Analysis (which is a pretty interesting topic).

For Topology, if you have read some of the analysis books above, I would say Munkres' book is nice and it has tons of examples. But try googling beginner topology books if you want to get into the subject sooner, I know I have seen a few stackexchange threads on this.

These are really the topics one needs to know to really dive into mathematics beyond rote computation. I'm sure there are more books out there but these come off my head at this moment.

This is a good one

Just keep in mind, you'll need a solid background in algebra to do algebraic number theory meaningfully

Knapp's book is good, but it is quite advanced and not self contained. It assumes a course in Lie groups, ("as in Chapter IV of Chevalley [1946]"). One of my favorite books, though.

I had a class as an undergrad that used Topics in Applied Abstract Algebra . It may be above the level you're looking for, but it covers some topics that may be appropriate- including crypto, balanced incomplete block designs, and wallpaper groups.

It does require a (relatively basic) understanding of what groups and rings are.