关于真增强幂图为无爪图的有限群

中文摘要 AI 辅助

设$G$为有限群。$G$的**增强幂图**记为$\u27e8(G)$，是顶点集为$G$的图，其中两个顶点$u$和$v$相邻当且仅当存在元素$w \u2208 G$使得$u$和$v$都属于$\u27e8w\u27e9$。$G$的**真增强幂图**记为$\u27e8^{**}(G)$，是$\u27e8(G)$由非支配顶点诱导的子图。\n本文的主要目标是研究真增强幂图为无爪图（即不包含同构于完全二部图$K_{1,3}$的诱导子图）的有限群。我们首先证明，$\u27e8(G)$是无爪图当且仅当$G$是循环群。$\u27e8(G)$的支配顶点集构成$G$中心的一个循环子群，即$G$的**循环化子**$\text{cyc}(G)$。这使得我们能够在$\u27e8^{**}(G)$为无爪图时，精确刻画$G/\text{cyc}(G)$的结构。若$G$可解但非幂零，则$G$是亚循环群，或者$G/\text{cyc}(G)$是Frobenius群或2-Frobenius群。若$G$不可解，则$G/\text{cyc}(G)$同构于$\text{PSL}(2,q)$或$\text{PGL}(2,q)$，这使得我们能够对真增强幂图为无爪图的不可解群给出完全分类。

英文摘要

Let $G$ be a finite group. The \emph{enhanced power graph} of $G$, denoted by $\mathcal{E}(G)$, is the graph with vertex set $G$ in which two vertices $u$ and $v$ are adjacent if and only if there exists an element $w \in G$ such that both $u$ and $v$ belong to $\langle w \rangle$. The \emph{proper enhanced power graph} of $G$, denoted by $\mathcal{E}^{**}(G)$, is the subgraph of $\mathcal{E}(G)$ induced by the non-dominating vertices. The main objective of this paper is to investigate finite groups whose proper enhanced power graph is claw-free, that is, contains no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. We first prove that $\mathcal{E}(G)$ is claw-free if and only if $G$ is cyclic. The set of dominating vertices of $\mathcal{E}(G)$ forms a cyclic subgroup of the center of $G$, namely the \emph{cyclicizer} $\cyc(G)$ of $G$. This allows us to give a precise description of the structure of $G/\cyc(G)$ when $\mathcal{E}^{**}(G)$ is claw-free. If $G$ is solvable but not nilpotent, then $G$ is metacyclic, or $G/\cyc(G)$ is either a Frobenius group or a $2$-Frobenius group. If $G$ is non-solvable, then $G/\cyc(G)$ is isomorphic to $\PSL(2,q)$ or $\PGL(2,q),$ and this allows us to give a complete classification of the non-solvable groups whose proper enhanced power graph is claw-free.

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15 pages