  In his 1934 paper, G.Birkhoff poses the problem of classifying pairs $(G,U)$,
where $G$ is an abelian group and $U\subset G$ a subgroup, up to automorphisms
of $G$. In general, Birkhoff's Problem is not considered feasible. In this
note, we fix a prime number $p$ and assume that $G$ is a direct sum of cyclic
$p$-groups and $U\subset G$ is a subgroup. Under the assumption that the factor
group $G/U$ is $p$-bounded, we show that each such pair $(G,U)$ has a unique
direct sum decomposition into pairs of type $(\mathbb Z/(p^n),\mathbb Z/(p^n))$
or $(\mathbb Z/(p^n), (p))$. On the other hand, if we assume $U$ is a
$p$-bounded subgroup of $G$ and in addition that $G/U$ is a direct sum of
cyclic $p$-groups, we show that each such pair $(G,U)$ has a unique direct sum
decomposition into pairs of type $(\mathbb Z/(p^n),0)$ or $(\mathbb Z/(p^n),
(p^{n-1}))$. We generalize the above results to modules over commutative
discrete valuation rings.
