A Unimodality Result in the Enumeration of Subgroups of a Finite Abelian Group

Abstract

The number of subgroups of order pk in an abelian group G of order pn is a polynomial in p, a\(k;p), determined by the type A of G. It is well known that ct\(k;p) = ot\(n -k;p). Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove that ot\(k;p), 0 < k < n, is a unimodal sequence of polynomials.