有限群的Chebotarev不变量的一个上界
An upper bound on the Chebotarev invariant of a finite group
摘要：
如果集合$\{g_1^{x_1}, \ldots, g_d^{x_d}\}$对$x_i \in G$的每个选择都生成$G$，则有限群$G$的子集$\{g_1, \ldots , g_d\}$总是生成$G$。$G$的Chebotarev不变量$C(G)$是随机变量$n$的期望值，该期望值最小，满足$n$随机选择的$G$元素总是生成$G$的要求。第一作者最近证明了对于某些绝对常数$β$，$C(G)\le β\sqrt{|G|}$.在本文中，我们表明，当$G$是可溶的，那么$β$是最多$5/3$。我们还表明，这是最好的可能。此外，我们表明，在一般情况下，对于每个$ε>0$存在一个常数$c_ε$，使$C(G)\le (1+ε)\sqrt{|G|}+c_ε$。
英文摘要：
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The first author recently showed that $C(G)\le β\sqrt{|G|}$ for some absolute constant $β$. In this paper we show that, when $G$ is soluble, then $β$ is at most $5/3$. We also show that this is best possible. Furthermore, we show that, in general, for each $ε>0$ there exists a constant $c_ε$ such that $C(G)\le (1+ε)\sqrt{|G|}+c_ε$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
