
Re: Showing group is Abelian
Posted:
Nov 30, 2012 2:25 PM


On Fri, 30 Nov 2012 14:18:46 +0000 (UTC), Michael Stemper <mstemper@walkabout.empros.com> wrote in <news:k9af86$m35$1@dontemail.me> in alt.math.undergrad:
> 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 handwavy.
> 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?
I don't offhand see a simpler approach. (3) isn't really a problem: once you've shown that an arbitrary product of m p's and n q's is equal to p^m q^n, show that the group is isomorphic to Z x Z. (Here m and n can be negative, a product of 3 p's, for instance, being (p^{1})^3.)
By the way, you might want to take a look at
<math.stackexchange.com>;
this would have been a perfect question for the site.
Brian

