The Chebotarev invariant for direct products of nonabelian finite simple groups

Jessica Anzanello, Andrea Lucchini, Gareth Tracey
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Abstract

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ 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 $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=\log{k}/\log{\alpha(G)}+O(1)$, where $\alpha$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(\log{k})$. We also derive sharp bounds on the expected number of generators for $G$.
非阿贝尔有限简单群直积的切博塔列夫不变式
如果 ${g_1^{x_1}, \ldots , g_d^{x_d}}$ 在 G$ 中每选择一个 x_i 都生成 $G$,那么有限群 $G$ 的子集 ${g_1, \ldots , g_d^{x_d}$ 不变地生成 $G$。$G$的切波塔列夫不变式$C(G)$是随机变量$n$的期望值,该期望值在随机选择$G$中的$n$元素不变地生成$G$的条件下是最小的。在本文中,我们证明了如果 $G$ 是非标注有限简单群,那么 $C(G)$ 是绝对有界的。更广义地说,我们证明了如果 $G$ 是 $k$ 非标注有限简单群的直接乘积,那么 $C(G)=\log{k}/\log{alpha(G)}+O(1)$, 其中 $\alpha$ 是一个不变量,完全由 $G$ 中因子的基元置换作用的邻接比例决定。根据波士顿-沙列夫猜想的证明,$C(G)=O(\log{k})$.我们还推导出$G$的预期生成数的尖锐边界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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