Prove the following proposition by invoking the Fundamental Theorem of Finite Abelian Groups.
Proposition: Let be a finite abelian group such that, for all positive integers dividing , contains at most elements satisfying . Prove that is cyclic.
Proof: By FTFGAG, we have for some integers such that for all . Suppose , and let be a prime divisor of . Now also divides . Now contains an element of order , and also contains an element of order . Now every element satisfies , and there are such elements, a contradiction. Thus , hence is cyclic.