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If we have a finite abelian group $G$ given as its invariant factor decomposition $G \cong C_{a_1} \times \dots \times C_{a_n}$ where $a_1 | \dots | a_n$, is the group $C_{a_1} \times \dots \times C_{a_1}$ ($n$ factors) a subgroup of $G$?

I believe it to be, but just want to confirm.

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    We have $C_{a_1}$ is clearly a subgroup of each $C_{a_k}$, then the product of subgroups is a subgroup of the product of the groups. Is this a valid argument? – 2017-02-06

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Let $g_i$ be the generators of the $C_i$ then the subgroup is generated by $g_1, g_2^{\frac{a_2}{a_1}}, \ldots, g_i^{\frac{a_i}{a_1}}, \ldots, g_n^{\frac{a_n}{a_1}}$.