An Introduction to Groups in Abstract Math

Date: 04/23/2008 at 09:37:31
From: John
Subject: abstract maths

I am writing this post to get some advice from you.  When I was in
school I was good at solving maths problems, not only routine types
but also Maths Olympiad types.

Please tell me your view of abstract concepts like groups.  How should
one go about learning it?   Should one mug up the definition?  On the
other hand there are too many definitions in abstract mathematics, how
should one form a logical connection and digest them?

I will be very thankful for your advice.

Date: 04/23/2008 at 16:17:24
From: Doctor Louis
Subject: Re: abstract maths

Dear John,

Abstract math is something deeply mystifying and is actually very
personal!  Your motivation for a definition may be completely
different than someone else's.

Let me take the example of groups:

To nearly everyone groups are defined as sets, together with an
associative binary operation such that 

  1) there exists an e for which for all g in G: ge=eg=g

  2) for every g in G there exists an h in G such that gh=hg=e

When I first saw this definition, it didn't seem intuitive to me; I
didn't really see the point in studying those objects.  So I started
to twist and turn the definition, and saw that those 2 conditions are
actually equivalent to saying: 

  for all x and y in G there is an a in G such 
  that xa=y and there is a b in G such that bx=y

which made much more sense to me, because studying the way elements
are connected seemed like the core of algebra.

All this is to say that I can't really help you, you have to figure
these things out for yourself.

But what I can do is recommend some books that have helped me a lot:

For groups, if you don't know anything about the subject, the best
introduction is John Fraleigh's _A First Course in Abstract Algebra_
(very complete and very step by step).

Another good source, is Serge Lang's _Algebra_.  Although he always
has the most elegant approach, his books tend to be very difficult, so
I would read that one once you know a bit about the subject.

And last but not least, Marshall Hall's _The Theory of Groups_ is a
classic, although again, I would start with Fraleigh.

Best of luck and don't hesitate to ask questions.

- Doctor Louis, The Math Forum
  http://mathforum.org/dr.math/