Hi
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?
i don't think anybody knows the answer! you probably had this in your mind that the answer is the lcm of the orders, but that is not necessarily true even for abelian groups. for example look at
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.