海森堡群的高-庄猜想

中文摘要 AI 辅助

设G为有限非阿贝尔群，G的小达文波特常数$\boldsymbol{\textsf d}(G)$是G上不存在乘积为1的子序列的最长序列长度，而G的高常数$E(G)$是使得G上长度不小于该值的任意序列都包含恰好长度为$|G|$的乘积为1的子序列的最小整数。庄和高提出的长期猜想断言，对每个有限非阿贝尔群G，有$E(G)=\boldsymbol{\textsf d}(G)+|G|$。设p为奇素数，$H_{p^3}=\boldsymbol{\text{UT}}_3(\boldsymbol{\text{F}}_p)$是阶为$p^3$、指数为p的海森堡群。Godara和Sarkar已证明阶为27、指数为3的非阿贝尔群满足庄-高等式，并询问对每个奇素数p，$H_{p^3}$是否也满足该等式。近期Volkmann证明$\boldsymbol{\textsf d}(H_{p^3})=3p-3$，本文确定了$H_{p^3}$的高常数，证明$E(H_{p^3})=\boldsymbol{\textsf d}(H_{p^3})+|H_{p^3}|=p^3+3p-3$。

英文摘要

Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb F_p)$ be the finite Heisenberg group over $\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\operatorname{UT}_3(\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order $p^3$, for every prime $p$.

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17 pages