# Textbook to study group theory as a part of Discrete Mathematics

I am a student from CS background. I have been following "Discrete Mathematics and its Applications" by Kenneth Rosen, though it is a good book, but it does not cover group theory. I would like to concentrate on the following topics and if required I am prepared to study more to build my foundations to understand the topics which I said shall be concentrating on. The topics are as follows:

Algebraic Structures and Morphism: Algebraic Structures with one Binary Operation, Semi Groups, Monoids, Groups, Congruence Relation and Quotient Structures, Free and Cyclic Monoids and Groups, Permutation Groups, Substructures, Normal Subgroups, Algebraic Structures with two Binary Operation, Rings, Integral Domain and Fields.

I would like to learn about a textbook which deals with them starting from introductory level. As I would be using the book for self study, I request the language of the book to be quite easy.

Is "Elements of Discrete Mathematics" by C.L. Liu a good choice?

• Rather than seek a Discrete Math book that also covers basic group theory (I guess your intent with Liu), why not supplement Rosen with a book focussed specifically on group theory? For example, A first course in Abstract Algebra by ‎Fraleigh. – Joseph O'Rourke Jun 11 at 0:54
• How is the language of Fraleigh is it comprehendible on the first read by a beginner?@Joseph – Abhishek Ghosh Jun 11 at 9:38
• Yes, Fraleigh is very clear for beginners, and has many easy exercises on which the self-learner can check themselves. It is a classic, in its 7th(!) edition. – Joseph O'Rourke Jun 11 at 11:22
• Fraleigh is quite approachable. I learned out of Hungerford's undergraduate text Abstract Algebra, and found that quite useful. Dummit and Foote is also a good reference to have around. Finally, if you really are only looking for something introductory, and you are primarily interested in groups, the MAA publication Visual Group Theory is quite nice. I'll note that I am answering in the comments because I think that the question itself is rather opinion-based, and phrased as a yes/no ("Is this book any good?"), neither of which is a great fit for this site. – Xander Henderson Jun 12 at 4:24
• @XanderHenderson, rather than closing, couldn't it be suited to this site simply by asking what books approach the union of discrete math and group theory from the desired point of view? (I agree with @‍JosephO'Rourke that I'm not sure why one would want this, but it seems like a reasonable thing to ask.) – LSpice Jun 16 at 16:35

The list of topics you want to study corresponds rather to abstract algebra than group theory. You did not say why you are interested specifically in group theory, but I believe that acquaintance with various algebraic structures in addition to groups would be beneficial for any person learning mathematics beyond school mathematics. Therefore, abstract algebra.

I wish to recommend

Gallian, Joseph. Contemporary Abstract Algebra. 9th ed., Cengage Learning, 2017.

It is suitable for beginners because it has many examples explained in great detail, sometimes even tediously. Also it contains many exercises. The author included some applications. The book's level of generality is low for my tastes, but it is okay for a beginner.

The phrase “discrete mathematics” is not a useful keyword for searching books on mathematics. As you can read in the Wikipedia, the scope of Discrete Mathematics is defined by what it does not contain (Analysis). There is no clear-cut set of notions that Discrete Mathematics is about. Its best definition is “mathematics for computer science” which, naturally, is a lot of things.

• did you mean the 9th edition? – Abhishek Ghosh Jun 11 at 20:58
• @AbhishekGhosh yes, thank you – beroal Jun 12 at 12:31

To repeat my comments: I can recommend Fraleigh's classic introduction. It is easy to read for beginners, with many exercises, from easy to difficult, on which self-learners could check themselves.

John B. Fraleigh A First Course in Abstract Algebra, 7th Edition. Pearson, 2002.

Because it's been around so long (1971?), there are many opportunities to find inexpensive used copies.

• Because it's been around so long (1971?) --- More like 1967, and in fact I have a copy of this edition (more precisely, the 3rd printing from May 1969), purchased new for $11.95 in 1975. According to my notes about this book, the 2nd, 3rd 4th, 5th 6th, 7th editions were published in 1976, 1982, 1989, 1994, 1999, 2002. Incidentally, the first 2 editions, but not the later editions, include 4 chapters (approximately 30 pages) that give a nicely written and very elementary introduction to homology groups and algebraic topology, this observation for others who might be interested, of course. – Dave L Renfro Jun 11 at 16:56 • @Abhishek Ghosh: You might want to consider one of the "applied abstract algebra" texts, such as Topics in Applied Abstract Algebra by Nagpaul/Jain (2005). I don't really have much advice on other possibilities, however, as the only reason I have this book is that a new copy was sent to me by the publishers in late 2004 or early 2005 for adoption consideration purposes, presumably due to my having taught an undergraduate abstract algebra course twice in the previous two years. – Dave L Renfro Jun 11 at 17:03 • @DaveLRenfro: The 7th edition of Fraleigh reintroduced the homology chapters. (I have the book on my bookshelf.) – J W Jun 11 at 17:20 • If you get this book, get used to the admonition to "never underestimate results that count something". Was this his go-to advice in early editions? – Nick C Jun 11 at 19:49 • @Nick C: Was this his go-to advice in early editions? --- It took me about 5 or 6 minutes, but I found the following in the 1967 1st edition, italicized on p. 93 (line$-7)\$: Never underestimate a theorem that counts something. Regarding shelf-life, Rudin's Principles of Mathematical Analysis (first published in 1953) is the oldest book I know of off-hand that gets widely recommended on the internet. Sure, Hardy's A Course of Pure Mathematics is also recommended a lot (first appeared in 1908), but not in the same way that Rudin's book is --- for current graduate school preparation. – Dave L Renfro Jun 11 at 20:52

I'll echo @XanderHenderson in the comments that Hungerford's Abstract Algebra: An Introduction is a really nice textbook. It's what I had as an undergraduate, and I'm currently re-reading it for maybe the third time. It's one of my favorite, well-written, clear texts.

Note that he's committed to a pedagogy that starts from the most familiar/concrete and gets subsequently more abstract, hence the sequence goes: Modular Arithmetic, then Rings, then Groups.

It's possible you just need the group theory, not the abstract algebra first. Would be good to look into what you need, before trying to satisfy it. The math types here will assume you need A before B. But for us scientists it's not always needed to roll how they do. As a practical example consider the books for chemists and spectroscopists on group theory (e.g. Cotton's). Very far from a "have to learn abstract algebra in order" approach. But give you the insights needed to understand why different shaped molecules have different numbers of vibrational modes (and consequent spectrums).

• Just a note that the OP mentions a CS background rather than chemistry. – J W Jun 12 at 8:19
• The OP explicitly specified topics outside group theory. – beroal Jun 12 at 12:34
• So, obviously, Cotton's Chemical Applications of Group Theory may not be the right thing for this dude. But it might be Madeupname's Computer Science Applications and Examples of Group Theory. (That's not a real book either. But a thought example. Now, I feel like I have to say that ahead of time or you will try to correct me with "I can't find that book".) ;-) – guest Jun 12 at 19:04
• It was just a note regarding the emphasis of your answer. I can see your point that you are using chemistry as an example. You could, however, have connected more explicitly with CS. All in the spirit of making answers better, albeit subjectively! – J W Jun 13 at 5:37
• By the way, I have neither upvoted nor downvoted your answer (at the time of writing this comment). I can see that you're trying to make a helpful point to the OP and if you were to edit to remove the generalization about "the math types here" and possibly say something about CS, I would consider upvoting. – J W Jun 14 at 11:38