I am looking to a reference to a solution of Exercise 13 (page 91) of L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, 1970. There is no solution to this exercise in a collection Exercises in Abelian Group Theory by Calugraenu at al.
The exercise is as folows: Let $A=A_1\oplus\dots\oplus A_k$, where $A_i$ is the direct sum of $m_i$ (a cardinal number, finite/infinite) of copies of $Z(p^i)$, $i=1,2,\dots,k$. Describe structures of subgroups and quotient groups of $A$.
My question is the extent of possible (or better, known) answer to this exercise.
Let $B$ be a subgroup of $A$. As $A$ is a bounded group, then both $B$ and $A/B$ are bounded, as well. By First Prufer theorem, $B$ and $A/B$ are direct sums of cyclic $p$-groups. Orders of elements in $B$ and $A/B$ do not exceed $p^k$, hence $B=B_1\oplus\dots\oplus B_k$ and $A/B=C_1\oplus\dots\oplus C_k$, where $B_i$ is a direct sum of say $n_i$ of copies of $Z(p^i)$, $C_i$ is a direct sum of say $l_i$ of copies of $Z(p^i)$ for $i=1,2,\dots,k$.
By Theorem 17.4 in mentioned L. Fuchs's book, cardinal numbers $m_1,\dots,m_k$ are invariants of $A$ - it means they do not depend on the direct sum representation of $A$. So that there would be nice to have some relations between (cardinal) numbers $m_1,\dots, m_k$ and $n_1,\dots, n_k$ (and between (cardinal) numbers $m_1,\dots, m_k$ and $l_1,\dots, l_k$ in the case of quotients).
I think that a slight generalization of the proof of mentioned Theorem 17.4 from L. Fuch's book can lead to the following (for subgroups):
$m_k\ge n_k$, $m_k+m_{k-1}\ge n_k+n_{k-1}$, $m_k+m_{k-1}+m_{k-2}\ge n_k+n_{k-1}+n_{k-2}$, ... ,$m_k+m_{k-1}+\dots+m_1\ge n_k+n_{k-1}+\dots+n_1$.
Is this what was expected as the answer to the Execrise 13 (for the case of structures of subgropus)?
Is this true also for quotient groups? - it means do the following relations hold
(*) $m_k\ge l_k$, $m_k+m_{k-1}\ge l_k+l_{k-1}$, $m_k+m_{k-1}+m_{k-2}\ge l_k+l_{k-1}+l_{k-2}$, ... ,$m_k+m_{k-1}+\dots+m_1\ge l_k+l_{k-1}+\dots+l_1$?
These relations (for subgroups and/or for quotient groups) can be viewed as generalization of similar relations for finite abelian case (see e.g. Subgroups and Quotient Groups of finite Abelian p-groups)
By an easy cardinal argument one can see that formulas (*) are true for the case when $A$ is a direct sum of $m_k$ (and $m_k$ is an infinite cardinal, it works for finite case as well, but that is not interesting just now) copies of a group $Z(p^k)$ (it means that $m_{k-1}=\dots=m_1=0$ for this group $A$).
Is there some "official" solution (I know that this term does not have much meaning) to this exercise? Can someone give a reference?
