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Let $$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group.

What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $G=\mathbb{Z}/3\times$G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8$, mathbb{Z}/8,$$then I can take$$H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8.$$Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism). Which are the subgroups H in G for which G/H is primary cyclic? 10 added 2 characters in body Let$$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$be any finite abelian group. What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8, then I can take H=3(\mathbb{Z}/3\times H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8. mathbb{Z}/8.$$ Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism). Which are the subgroups$H$in$G$for which$G/H$is primary cyclic? 9 Restored the title. Fixed LaTeX. # AretheseallthesubgroupsSubgroups of a finite abelian group? Let$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n} be any finite abelian group.

What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8$, then I can take $H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8$. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).

Which are the subgroups $H$ in $G$ for which $G/H$ is primary cyclic?

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Post Reopened by Mark Sapir, Qiaochu Yuan, HW, Anton Geraschenko♦♦
Post Closed as "too localized" by Franz Lemmermeyer, Pete L. Clark, José Figueroa-O'Farrill, HW, Mariano Suárez-Alvarez
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