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

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