Abelian Group Structures

Question

How can I determine all the subgroups of a commutative group, write the Hasse diagram, using Frobenius-Stickelberger Theorem and the isomorphism to $\mathbb{Z}_m$ of a cyclic group? In particular, for example, if $|G|=n$, and $n=pq$ with $p$ and $q$ prime, how can I decide if $G$ is isomorphic to $\mathbb{Z}_n$ or to $\mathbb{Z}_p\times\mathbb{Z}_q$ (I mean without trying to think and build an isomorphism, but using some theorem or anything else) ?

Answer 1

the best way to do this is to think that the first factor in the decomposition of $G$ under cyclic groups is of maximum order, more precisely its order is the exponent of $G$, to joust at Cauchy result who assure that for all prime $p$ dividing the order of $G$, it also divides the exponent of this group. As noted in the comments, if $p$ and $q$ are distinct then it is cyclic group of order $pq$ (for abelian group, say it is cyclic is equivalent to his order equal to his exponent) so isomorphic to $\Bbb{Z}/pq\Bbb{Z}$, particularly $\Bbb{Z}/p\Bbb{Z}\times\Bbb{Z}/q\Bbb{Z}$ isomorphic to $\Bbb{Z}/pq\Bbb{Z}$; in the case of $p = q$ are two possibility for the structure of G, depending in the exponent of $G$ is $p$ or $ p^2$ and $G$ is respectively isomorphic to $\Bbb{Z}/p\Bbb{Z}\times\Bbb{Z}/p\Bbb{Z}$ or $\Bbb{Z}/p^2\Bbb{Z}$