Finite Abelian Groups
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.
Originally Posted by tonio
Thank you for your response. In your example, for the group G of order , isn't there are twelve possible ways we could express the direct product? For instance, can G be written as as well? So, in this case, isn't its maximal cyclic subgroup is ? How do we know which of these twelve choices will give us the maximal cyclic subgroup? Also, you mentioned that for my problem, it has a sbgp of order prq which must be cyclic and of size 3, and it's divided by all the three primes dividing the order of G. But how do we know that subgroup of order prq is the largest subgroup? How about subgroup with order ? Isn't it greater than prq? Finally, how do we know all the three primes divided 3? Since 3 is a prime number, its divisors must be 1 and itslef only, right? Appreciate your clarification on this. Thank you, Tonio.
Originally Posted by anlys