# Looking for a good abstract book

1. May 13, 2004

### AndyCiep

Hi, I was just hoping for some suggestions for a good book to learn abstract algebra with. Specifically, I'd like a book with a lot of real-world references, or practical application problems. Consider me a beginner. I've actually already taken abstract (just finished it this past semester) but, unlike calculus, statistics, and physics.. I had a difficult time grasping "how any of this matters" (so to speak ). For me, it all seemed like a jumble of new definitions and concepts without any allusion as to why I should want to know any of it. As a result, I think it became a tad difficult for me to follow and I didn't do as well as I'd have liked.

Any suggestions for a good book(s) that might nurture my somewhat stubborn "tell me why I should care" attitude?

thanks everybody
_Andrew

2. May 13, 2004

### matt grime

Jacobsen's Algebra is good but expensive, Representations and Characters of Groups by Liebeck and James.

If you want applications then the standard ones are:

without ring theory you wouldn't have the encryption techniques on the internet. indeed cryptography is a generally useful example, currently I believe that the security services think that elliptic curves may offer more codes.

then we can use it in chemistry, where the normal modes of vibration of atoms in molecules can be found using representations of the symmetry group.

advanced particle physics basically IS the study of lie algebras; and don't forget gauge groups, and category theory, and such seemiingly bizarre statements as fermions correspond to tracial elements in the infinite tensor algebra of $$M_2\otimes M_2\otimes M_2\ldots$$

3. May 14, 2004

### slyboy

It depends a bit on what you mean by "real world applications".

If you like Computer Science applications then you might try supplementing your reading with:

Applied Abstract Algebra
by Rudolf Lil, Rudolf Lidl and Gunter Pilz

It assumes you have already studies a first abstract algebra course, but it is nice because it shows applications of many branches of algebra. Most other books at this level only look at the applications of groups. Speaking of which, if you want to learn how to solve puzzles then try:

Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys
by David Joyner

This one develops the theory from first principles.

If you prefer the hard sciences then you might try picking up a book on crystalography, which makes a lot of use of finite group theory.

The main physics applications use Lie groups, so you will have to get through a topology course as well before you can study them rigorously. Before then, you could look at:

Groups, Representations and Physics
by H. F. Jones

but it is liable to give serious mathematicians a heart attack due to its rather informal presentation.

4. Jul 14, 2004

### hawthorn

Another good applied text is Applied Modern Algebra, by Larry Dornhoff and Franz Hohn. It's currently out of print but you can get used copies at abebooks on the web.

This book is geared toward advanced undergrads in Electrical and Computer Engineering as well as Computer Science. This book contains a wealth of applied material on Finite State Machines, Lattices, Graph Theory, Coding Theory, as well as the usual Group Theory, Ring Theory, etc.

I completely understand the frustration of getting hold of the wrong book in studying these abstract algebraic structures. Without some appreciation for how this can be applied in the real world, it just seems rather pointless.

5. Jul 19, 2004

### gravenewworld

The best book for algebra hands down is Gallian's Contemporary Abstract algebra, it is very clear and concise. It also has applications to real world problems.

6. Jul 19, 2004

### plover

You might look at Michael Artin's Algebra.

7. Jul 19, 2004

### Tom Mattson

Staff Emeritus
I took the course from a book by a guy named Fraleigh (A First Course in Abstract Algebra, I think it was). I really liked it, and still refer to it.

8. Jul 20, 2004

### phoenixthoth

i recommend the two a.a. books by herstein.