关于其幂图无爪的有限群
On finite groups whose power graph is claw-free
AI总结：
本文探讨了简化幂图无爪的有限群，证明了此类群要么是可解群要么是基座同构于 $PSL(2,q)$ 的几乎单群，且其阶最多被 5 个不同素数整除。
摘要：
如果一个图不包含同构于完全二部图 $K_{1, 3}$ 的导出子图，则称该图为无爪的。群 $G$ 的无向幂图以 $G$ 的元素为顶点，当两个循环子群 $\langle g_1\rangle, \langle g_2\rangle$ 中的一个包含在另一个中时，$g_1$ 和 $g_2$ 之间有一条边，记作 $P(G)$。简化幂图记作 $P^*(G)$，是 $P(G)$ 由非单位元诱导的子图。本文的主要目的是探索简化幂图无爪的有限群。特别地，我们证明如果 $P^*(G)$ 无爪，则 $G$ 要么是可解群，要么是几乎单群。在第二种情况下，对于合适的 $q$ 的选择，$G$ 的基座同构于 $PSL(2,q)$。最后我们证明，如果 $P^*(G)$ 无爪，则 $G$ 的阶最多被 5 个不同的素数整除。
英文摘要：
A graph is called claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph $K_{1, 3}$. The undirected power graph of a group $G$ has vertices the elements of $G$, with an edge between $g_1$ and $g_2$ if one of the two cyclic subgroups $\langle g_1\rangle, \langle g_2\rangle$ is contained in the other. It is denoted by $P(G)$. The reduced power graph, denoted by $P^*(G),$ is the subgraph of $P(G)$ induced by the non-identity elements. The main purpose of this paper is to explore the finite groups whose reduced power graph is claw-free. In particular we prove that if $P^*(G)$ is claw-free, then either $G$ is solvable or $G$ is an almost simple group. In the second case the socle of $G$ is isomorphic to $PSL(2,q)$ for suitable choices of $q$. Finally we prove that if $P^*(G)$ is claw-free, then the order of $G$ is divisible by at most 5 different primes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
