Direct product of subgroups of factors in direct product decomposition into cyclic groups

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Definition

Suppose $G$ is a finitely generated abelian group. A subgroup $H$ of $G$ is termed a direct product of subgroups of factors in direct product decomposition into cyclic groups if there exists a decomposition of $G$ as an internal direct product of cyclic groups:

$G = G_1 \times G_2 \times \dots \times G_n$

and subgroups $H_i \le G_i$ such that:

$H = H_1 \times H_2 \times \dots \times H_n$