A unimodality result in the enumeration of subgroups of a finite abelian group

Abstract

The number of subgroups of order p k {p^k} in an abelian group G G of order p n {p^n} is a polynomial in p , α ⋋ ( k ; p ) p,{\alpha _ \leftthreetimes }(k;p) , determined by the type λ \lambda of G G . It is well known that α ⋋ ( k ; p ) = α ⋋ ( n − k ; p ) {\alpha _ \leftthreetimes }(k;p) = {\alpha _ \leftthreetimes }(n - k;p) . Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove that α ⋋ ( k ; p ) {\alpha _ \leftthreetimes }(k;p) , is a unimodal sequence of polynomials. That is, for 1 ≤ k ≤ n / 2 , α λ ( k ; p ) − α λ ( k − 1 ; p ) 1 \leq k \leq n/2,{\alpha _\lambda }(k;p) - {\alpha _\lambda }(k - 1;p) is a polynomial in p p with nonnegative coefficients.