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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A unimodality result in the enumeration of subgroups of a finite abelian group
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by Lynne M. Butler PDF
Proc. Amer. Math. Soc. 101 (1987), 771-775 Request permission

Abstract:

The number of subgroups of order ${p^k}$ in an abelian group $G$ of order ${p^n}$ is a polynomial in $p,{\alpha _ \leftthreetimes }(k;p)$, determined by the type $\lambda$ of $G$. It is well known that ${\alpha _ \leftthreetimes }(k;p) = {\alpha _ \leftthreetimes }(n - k;p)$. Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove that ${\alpha _ \leftthreetimes }(k;p)$, is a unimodal sequence of polynomials. That is, for $1 \leq k \leq n/2,{\alpha _\lambda }(k;p) - {\alpha _\lambda }(k - 1;p)$ is a polynomial in $p$ with nonnegative coefficients.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 101 (1987), 771-775
  • MSC: Primary 05A15; Secondary 20D60, 20K01
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0911049-8
  • MathSciNet review: 911049