Discrete valuation rings, partitions and $p$-groups I

A finite abelian $p$-group having an automorphism $x$ such that $1+\ldots+x^{p-1}=0$, can be viewed as a module over an appropriate discrete valuation ring $\mathcal{O}$ containing $\mathbb{Z}_p$ (the ring of $p$-adic integer). This yields the natural problem of comparing the invariants of $A$ as a $\mathbb{Z}_p$-module to its invariants as an $\mathcal{O}$-module. We solve the latter problem in a more general context, and give some applications to the structure of some $p$-groups and their automorphisms.