Let G be a finite Abelian group of order where p, q, and r are distinct primes. Define "size" as the number of prime factors (For example, and have size 3, but have the size 2, even though might be larger than both and )
Question: What is the minimum size for "the" largest cyclic subgroup of G and which primes divide the size of this subgroup? Explain your answer.
I am not too sure how to do this problem. By the Fundamental Theorem of Finite Abelian Groups, there are 12 possible choices of groups that G is isomorphic to, right? I am really confused on how to find "the" largest cyclic group of G here. Any help would be greatly appreciated. Thank you in advance.