由互指数子群生成有限群
Generation of finite groups from subgroups of coprime index
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中文总结 AI 辅助
该研究解决Kourovka问题21.87，证明有限群$G$若有子群族满足各子群最小生成集大小不超$d$且指数最大公约数为1，则$G$的最小生成集大小不超$d+1$，通过归约反例等方法完成证明。
中文摘要 AI 辅助
设$d(G)$表示有限群$G$的最小生成集大小。我们证明：若$G$有子群族$\boldsymbol{\textit{H}}$，满足对每个$H \boldsymbol{\textit{H}}$，$d(H)\boldsymbol{\textit{d}}$，且$\boldsymbol{\textit{G:H}}:H \boldsymbol{\textit{H}}$的最大公约数为1，则$d(G)\boldsymbol{\textit{d+1}}$。这对Kourovka问题21.87给出肯定回答。证明将极小反例归约为具有非阿贝尔根基的临界基于根基的幂。精确的根基重数公式和条件生成的统一下界，为根基因子数量提供下界。含Sylow 2-子群的子群，通过有限单群Sylow 2-子群的逐点中心化子估计，给出矛盾的上界。
英文摘要
Let $d(G)$ denote the least size of a generating set of a finite group $G$. We prove that if $G$ has a family $\mathcal H$ of subgroups such that $d(H)\leq d$ for every $H\in\mathcal H$ and $\gcd\{\lvert G:H\rvert:H\in\mathcal H\}=1$, then $d(G)\leq d+1$. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow $2$-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow $2$-subgroups of finite simple groups.
