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Topic: Showing group is Abelian
Replies: 6   Last Post: Dec 3, 2012 11:49 AM

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Michael Stemper

Posts: 671
Registered: 6/26/08
Showing group is Abelian
Posted: Nov 30, 2012 9:18 AM
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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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