AI 中文总结
该研究证实了有限非阿贝尔群的Erdős-Ginzburg-Ziv常数与短乘积一常数的等式成立，并确定了相关广义常数，针对具有最小素因子指数循环子群的有限非阿贝尔群。
AI 中文摘要
设$G$为有限群，$\text{exp}(G)$表示其指数。Erdős-Ginzburg-Ziv常数$s(G)$是迫使存在长度为$\text{exp}(G)$的乘积一子序列的最小整数，短乘积一常数$\text{η}(G)$是迫使存在长度不超过$\text{exp}(G)$的非空乘积一子序列的最小整数。有限阿贝尔群上关于Erdős-Ginzburg-Ziv常数的一个猜想（W. Gao，《关于受限大小的零和子序列 II》，《离散数学》2003年）的自然非阿贝尔扩展预测$s(G)=\text{η}(G)+\text{exp}(G)-1$。我们对每个具有指数为$p$的循环子群的有限非阿贝尔群$G$证实了该等式，其中$p$是$|G|$的最小素因子。作为进一步结果，我们确定了这类群的所有广义Erdős-Ginzburg-Ziv常数$s_{m\text{exp}(G)}(G)$。
英文摘要
Let $G$ be a finite group, and let $\exp(G)$ denote its exponent. The Erdős-Ginzburg-Ziv constant $s(G)$ is the least integer forcing a product-one subsequence of length $\exp(G)$, while the short product-one constant $η(G)$ is the least integer forcing a nonempty product-one subsequence of length at most $\exp(G)$. The natural nonabelian extension of a conjecture [W. Gao, \emph{On zero-sum subsequences of restricted size II}, Discrete Math. 2003] on the Erdős-Ginzburg-Ziv constant in finite abelian groups predicts that $s(G)=η(G)+\exp(G)-1.$ We confirm this equality for every finite nonabelian group $G$ having a cyclic subgroup of index $p$, where $p$ is the smallest prime divisor of $|G|$. As further consequences, we determine all generalized Erdős-Ginzburg-Ziv constants $s_{m\exp(G)}(G)$ for this family of groups.
Comments16 pages
