Abstract :
Let q be a prime number. The number of subgroups of order qk in an abelian group G of order qn and type λ is a polynomial in q, [kλ′]q. In 1987, Lynne Butler showed that the first difference, [kλ′] − [k − 1λ′], has nonnegative coefficients as a polynomial in q, when 2k ⩽ |λ|. We generalize the first difference to the rth difference, and give conditions for the nonnegativity of its coefficients.
