Let $G$ be a finite abelian group and $A\subseteq G$. For $n\in G$, denote by $r_{A}(n)$ the number of ordered pairs $(a_{1},a_{2})\in A^{2}$ such that $a_{1}+a_{2}=n$. Among other things, we prove that for any odd number $t\geq 3$, it is not possible to partition $G$ into $t$ disjoint sets $A_{1},A_{2},\dots ,A_{t}$ with $r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.

For a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of with ramification index e. The orders are given explicitly when G is a p-group of order p or p 2. An example is given when G is not abelian.

We solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.