What Should I Learn Before Studying Group Theory?

[zooxanthellae]

What concepts should I be familiar with in order to get anything meaningful out of a textbook on group theory? I've read a few articles that talk about a few of group theory's aims and subjects, and it's enough to pique my interest.

I've taken first-year Calculus and a bit of Analysis (the 160s Calc sequence at UChicago if that helps).

I'm assuming I do not have a strong enough math background. What, then, should I look at first? The plan is to learn it myself as a small hobby over summer break.

Thank you!

 

[snipez90]

Oh hey I sort of recognize your username.

You're actually probably prepared for learning group theory. You still have to take the analysis sequence, but that won't have much direct bearing on learning group theory.

Although the difficulty of 160's varies with the instructor, if you understand Spivak thoroughly and are at least tackling the harder problems, you should be fine.

Anyways, 163 isn't going to be much trouble if you've already done well in 161 and 162, so if you're interested in tackling group theory now PM me directly.

 

[doodle_sack]

Other than Set Theory & Functional Equations you only need some "mathematical maturity" to understand group theory. Having some knowledge in matrix operations wouldn't hurt though, but you can learn that on the way.

 

[mal4mac]

Try "A Course in Group Theory" by John F. Humphreys. It's aimed at someone at about your level.

 

[PhDorBust]

Other than Set Theory & Functional Equations you only need some "mathematical maturity" to understand group theory. Having some knowledge in matrix operations wouldn't hurt though, but you can learn that on the way.

Only very basic set theory... and what do you need to know functional equations for? Most introductory books will only require mathematical maturity.

 

[zpconn]

If you have no mathematical background, pick up "Abstract Algebra" by Herstein. It has no prerequisites. If you know some linear algebra, pick up Artin instead.

 

[Sankaku]

Although there are many solid books, this is the best intro to abstract algebra I have ever found:

A Book of Abstract Algebra
by Charles C Pinter
http://www.bookdepository.com/book/9780486474175/Book-of-Abstract-Algebra

It explains everything very clearly and the bonus is that it is incredibly inexpensive.

 

[hitmeoff]

Fraleigh's A First Course in Abstract Algebra is probably the clearest, easiest textbook introductions to Groups (as well as Rings, Fields, Galois THeory).

Group Theory is very different from other math. Technically, you really don't even need to know Calculus, all you REALLY need is basic arithmetic and algebra. But when people talk about "maturity" they mean the ability to read, understand and do rigorous proofs.

Alot of the "machinery" you do in group theory isn't much more complicated than basic algebraic manipulations, but at this point in your math career you are less interested in "calculating" and are more interested in "proving", "showing" etc. A lot of your basic "knowledge" about algebraic operations that you take for granted, you can't in group theory (and the rest of abstract algebra).

For example:

a*b = b*a is NOT a given (you should actually have seen this in linear algebra where matrices are not, in general commutative under multiplication). So when you prove stuff, you really have to know your theorems and definitions well and do only what those theorems and definitions say you can do, and nothing more.

Again, for the above example, if your proof of some problem NEEDS or SEEMS to need a*b = b*a, then you have to be sure that your algebraic structure guarantees that (it may not).

Group theory is often the hardest class a math major will take, not because DOING it is hard, but rather most people just are NOT used to THINKING about math in this way (most people have a ton of calculation experience and maybe a smidgen of proof experience).

Some courses that may prepare you for the type of thinking involved in Group Theory: Elementary Number Theory, junior level Linear Algebra, Modern Geometry. Those are given in level of difficulty (in my opinion) from Hardest to Easiest. What these course can provide is a "nicer" introduction to rigorous proofs and abstract thinking. They deal with structures and operations you probably have seen before, but now you develop it rigorously (same with abstract algebra, basic elementary algebra is derived from abstract algebra, but group theory tends to go off into areas of algebra you have definitely never seen before).

TO make a long story, still long: Group Theory can be very hard and frustrating at first, it'll probably require a lot time effort and a lot of hair being ripped out of your head; but if you work at it, give it a serious effort and (and this is VERY IMPORTANT) as for HELP from your Prof or TA, then you can do just fine. Its not impossible, just different. Go into it knowing its going to be something very different from Calc, Elementary Linear Algebra, Trig, etc...and go into it willing to learn this new type of thinking.

I can guarantee you that a semester or two of abstract algebra will make you into a much much better mathematician.