素幂阶的消失元素及其类大小性质

中文摘要 AI 辅助

通过对共轭类和特征标度数施加算术条件来研究群的结构，已取得若干有趣结果并提出了公开问题。受此工作的启发，Dolfi和Lucido在文献\ucf10{MR1826493}中为群引入了一个性质：对素数$p,q$，若群$G$中每个$p'$-元素都有$q'$-类大小，则称$G$具有性质$P(p,q)$，他们得到了当$G$满足该性质时$G$及其一些子群结构的若干结果。我们引入上述性质的消失类似性质：对素数$p \neq q$，若有限群$G$中每个素幂阶的消失$p'$-元素的共轭类大小都不被$q$整除，则称$G$具有性质$P_v(p,q)$。我们证明，不存在有限单群满足对整除$|G|$的素数$p \neq q$的性质$P_v(p,q)$；利用该结果可证明，若有限群$G$满足$p \neq q$且$p>2$的性质$P_v(p,q)$，则$O^{q'}(G)$（由$G$的所有西罗$q$-子群生成的子群）是可解的，这在更弱的条件下推广了Dolfi和Lucido的结果。

英文摘要

Study of the structure of groups by variation in the arithmetic conditions on conjugacy classes and character degrees has produced several interesting results and open problems. In continuation of such work, Dolfi and Lucido in \cite{MR1826493} introduced a property for groups. For primes $p,q$, a group $G$ is said to have property $P(p,q)$ if every $p'$-element in $G$ has $q'$-class size. They obtained several results on the structure of $G$ and of some subgroups when $G$ satisfies the property $P(p,q)$. Motivated by this work, we introduce a vanishing analogue of the above property: for primes $p \neq q$, a finite group $G$ is said to have the property $P_v(p,q)$ if every vanishing $p'$-element of prime power order in $G$ has conjugacy class size not divisible by $q$. We show that no finite simple group satisfies the property $P_v(p,q)$ for primes $p\neq q$ dividing $|G|$. We use this result to show that if a finite group $G$ satisfies the property $P_v(p,q)$ with $p \neq q$ and $p > 2$, then $O^{q'}(G)$ (subgroup generated by all Sylow $q$-subgroups of $G$) is solvable. This generalises a result of Dolfi and Lucido under weaker conditions.

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18 pages