论文标题
有限群体的Chebotarev不变的上限
An upper bound on the Chebotarev invariant of a finite group
论文作者
论文摘要
如果设置$ \ {g_1^{x_1}，\ ldots，g_d^{x_d} {x_d} $ g $ g $ x_i \ x__i \。 $ g $的Chebotarev不变性$ c（g）$ of $ g $是随机变量$ n $的预期价值，这是最小的要求，要求$ n $随机选择$ g $的元素总是会产生$ g $。第一作者最近表明，对于某些绝对常数$β$，$ c（g）\leβ\ sqrt {| g |} $。在本文中，我们表明，当$ g $可溶时，$β$最多为$ 5/3 $。我们还表明这是最好的。此外，我们表明，对于每个$ε> 0 $，存在一个常数$C_ε$，因此$ c（g）\ le（1+ε）\ sqrt {| g |}+c_ε$。
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 $C(G)\le (1+ε)\sqrt{|G|}+c_ε$.
