Abelian \(p\)-groups with a fixed elementary subgroup or with a fixed elementary quotient

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.