## 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.## References

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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