互素作用与不动点子群上非零的特征标

中文摘要 AI 辅助

设有限群 $A$ 互素地作用在有限群 $G$ 上，并令 $C=C_G(A)$。Burnside 的一个经典定理断言，一个群的非零特征标恰好是线性特征标，Navarro 在第 21 届 Kourovka 笔记的问题 21.100 中询问互素类比是否成立：具有 $\chi_C$ 处处非零的 $A$-不变 $\chi\in\Irr(G)$ 的个数是否总是 $|C/C'|$？我们对此给出否定回答。对于阶为 $21$ 的循环群 $A$ 作用在 $G=B\rtimes V$ 上，其中 $V=\F_4^2\oplus\F_8$，$B$ 是 $V$ 上的布尔函数群，不动子群 $C\cong C_2^{12}$ 承载全部 $4096$ 个不变特征标，而其中只有 $1728$ 个在 $C$ 上处处非零。由于 $C$ 是交换群，同样的例子反驳了 Navarro 问题列表中的问题 6.3，以及与之对应的 Isaacs 的 head 特征标的相关陈述；转而考虑 $G\rtimes A$ 则反驳了问题 6.7，即 Isaacs 报告称有大量计算证据支持的一个猜想。该构造仅需 $V$ 具有一个大小为一半的 $A$-稳定子集。这对无穷多对 $(A,V)$ 成立，且对任何维数低于 $7$ 的都不成立，因此该例子在所有奇数阶算子群中是最小的。与证明无关的精确机器验证随论文一同提供。

英文摘要

Let a finite group $A$ act coprimely on a finite group $G$ and put $C=C_G(A)$. A classical theorem of Burnside asserts that the irreducible characters of a group vanishing nowhere are exactly the linear ones, and Navarro asked in Problem 21.100 of the 21st Kourovka Notebook whether the coprime analogue holds: is the number of $A$-invariant $χ\in\Irr(G)$ with $χ_C$ nowhere zero always $|C/C'|$? We answer this negatively. For $A$ cyclic of order $21$ acting on $G=B\rtimes V$, where $V=\F_4^2\oplus\F_8$ and $B$ is the group of Boolean functions on $V$, the fixed subgroup $C\cong C_2^{12}$ carries all $4096$ invariant characters while only $1728$ of them are nowhere zero on $C$. Because $C$ is abelian the same example refutes Problem 6.3 of Navarro's problem list, and with it the corresponding statement about the head characters of Isaacs; passing to $G\rtimes A$ refutes Problem 6.7, a conjecture Isaacs reports is supported by abundant computational evidence. The construction needs only that $V$ have an $A$-stable subset of half its size. This holds for infinitely many pairs $(A,V)$, and for none of dimension below $7$, so the example is minimal over all operator groups of odd order. Exact machine verifications, independent of the proofs, accompany the paper.

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