格卢克猜想的一个证明
A Proof of Gluck's Conjecture
AI总结：
该研究证明了有限可迁置换群与有限可解群的相关不等式，进而解决了自1985年以来悬而未决的格卢克猜想。
AI中文摘要：
对于有限群H，令ν(H)表示H的幂零子群的最大阶。我们证明，有限集合Ω上的每个有限可迁置换群P，都存在子集Δ⊆Ω使得|P:P_Δ|≥ν(P_Δ)。我们还证明，若有限可解群H忠实且完全可约地作用在有限模V上，则存在x∈V满足|H:H_x|≥ν(H_x)。由此，我们解决了自1985年以来悬而未决的格卢克著名猜想：每个有限可解群G满足|G:𝐅(G)|≤b(G)²，其中𝐅(G)是G的最大正规幂零子群，b(G)是G的不可约复特征标的最大次数。
英文摘要：
For a finite group $H$, let $ν(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $Ω$ has a subset $Δ\subseteqΩ$ such that $|P:P_Δ|\geν(P_Δ)$. We also prove that if a finite solvable group $H$ acts faithfully and completely reducibly on a finite module $V$, then some $x\in V$ satisfies $|H:H_x|\geν(H_x)$. Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group $G$ satisfies $|G:\mathbf{F}(G)|\le b(G)^2$, where $\mathbf{F}(G)$ is the largest normal nilpotent subgroup of $G$ and $b(G)$ is the largest degree of an irreducible complex character of $G$.
