Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient

Abstract

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 an elementary abelian $p$-group, we show that the pair $(G,U)$ always has a direct sum decomposition into pairs of type $(\mathbb{Z}/(p^n),\mathbb{Z}/(p^n))$ or $(\mathbb{Z}/(p^n),(p))$. Surprisingly, in the dual situation we need an additional condition. If we assume that $U$ itself is an elementary subgroup of $G$, then we show that the pair $(G,U)$ has a direct sum decomposition into pairs of type $(\mathbb{Z}/(p^n),0)$ or $(\mathbb{Z}/(p^n),(p^{n-1}))$ if and only if $G/U$ is a~direct sum of cyclic $p$-groups. We generalize the above results to modules over commutative discrete valuation rings.