Hint: In a group, let be two elements of finite order which commute, what is the order of ?
For the non-abelian counterexample, you can take a look at .
Let a be an element of maximum order from a finite Abelian group G. Prove that for any element b, |b| divides |a| (order of b divides order of a). Show by example that this need not be true for finite non-Abelian Groups.
I'am really stuck on this one. Any help would be great
If the order of g=n and the order of f=m, then the order of gf would divide m*n. So if an element does not divide the element of maximum order in a finite Abelian group G, the order of fg>f when f is the maximal order element, this would result in a contradiction. Is this right?
the case anyway, what wutang needs here is this fact that for any two elements in a finite abelian group, there exists an element such that see here.