Enumeration of Certain Subgroups of Abelian p-Groups

Abstract

The number of distinct types of Abelian group of prime-power order pn is equal to the number of partitions of n. Let (ρ) = (ρ1, ρ2, …, ρr) be a partition of n and let (μ) = (μ1, μ2, …, μs) be a partition of m, with ρ1≧ρ2≧…≧ρr and μ1≧μ2≧…≧μs, ρi≧μi, r≧s, n>m. The number of subgroups of type μ in an Abelian p-group of type (ρ) is a function of the two partitions (μ) and p, and has been determined as a polynomial in p with integer coefficients by Yeh (1), Delsarte (2) and Kinosita (3). Their results differ in form but are equivalent.