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?
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