Date: Nov 30, 2012 9:18 AM
Author: Michael Stemper
Subject: Showing group is Abelian
I'm currently on a problem in Pinter's _A Book of Abstract Algebra_, in
which
Date: Nov 30, 2012 9:18 AM
Author: Michael Stemper
Subject: Showing group is Abelian
I'm currently on a problem in Pinter's _A Book of Abstract Algebra_, in
which the student is supposed to prove that the (sub)group generated by
two elements a and b, such that ab=ba, is Abelian.
I have an outline of such a proof in my head:
1. Show that if xy = yx then (x^-1)y = y(x^-1). This is pretty simple.
2. Use induction to show that if p and q commute, then any product of
m p's and n q's is equal to any other, regardless of order.
3. Combine these two facts to show the desired result.
However, this seems quite messy. I'm also wary that what I do for the
third part might end up too hand-wavy.
Is there a simpler approach that I'm overlooking, or do I need to just
dive in and go through all of the details of what I've outlined?
Michael F. Stemper
If this is our corporate opinion, you will be billed for it.