Let G be a finite abelian group. Then by structure theorem $G=Z_{n_1}×Z_{n_2}×Z_{n_3}×...×Z_{n_s}$
where $s\ge 1,n_{i+1}|n_i$ for $1\le i \le s-1$
Now let $H\le G$,again by the structure theorem
$H=Z_{m_1}×Z_{m_2}×Z_{m_3}×...×Z_{m_t} $
where $t\ge 1,m_{i+1}|m_i$ for $1\le i \le t-1$
My question is whether there is some sort of relationship between the structures of $H$ and $G$ ? I mean $t\le s$ and as such..
My intuition says yes though I am unable to proceed in any direction . I am also aware of the fact that subgroups of direct product may not be direct product of subgroups. Does it have some implication in this question ?.I have looked everywhere but in vain.
It will be very helpful if one can give me a proof or at least a sketch of proof with a bit of theoretical discussion since I am mostly self-studying.
Thanks and regard to you !!