On combinatorial properties of Gruenberg–Kegel graphs of finite groups

Monatshefte für Mathematik Pub Date : 2024-08-20 DOI:10.1007/s00605-024-02005-6

Mingzhu Chen, Ilya Gorshkov, Natalia V. Maslova, Nanying Yang

引用次数: 0

Abstract

If G is a finite group, then the spectrum $\omega (G)$ is the set of all element orders of G. The prime spectrum $\pi (G)$ is the set of all primes belonging to $\omega (G)$. A simple graph $\Gamma (G)$ whose vertex set is $\pi (G)$ and in which two distinct vertices r and s are adjacent if and only if $rs \in \omega (G)$ is called the Gruenberg–Kegel graph or the prime graph of G. In this paper, we prove that if G is a group of even order, then the set of vertices which are non-adjacent to 2 in $\Gamma (G)$ forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

论有限群的格伦伯格-凯格尔图的组合特性

如果 G 是一个有限群，那么谱 $\omega (G)$ 是 G 的所有元素阶的集合。素数谱 $\pi (G)$ 是属于 $\omega (G)$ 的所有素数的集合。一个简单的图（$\Gamma (G)$）的顶点集合是（$\pi (G)$），并且其中两个不同的顶点 r 和 s 相邻，当且仅当（$rs \in \omega (G)$）是且仅当（$rs \in \omega (G)$），这个图被称为格伦伯格-凯格尔图或 G 的素数图。在本文中，我们证明了如果 G 是偶数阶群，那么在 $\Gamma (G)$ 中与 2 不相邻的顶点集合构成了小群的联合。此外，我们还确定了强规则图何时与有限群的格伦伯格-凯格尔图同构。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Monatshefte für Mathematik

自引率

0.00%

发文量

期刊最新文献

On combinatorial properties of Gruenberg–Kegel graphs of finite groups Sparse bounds for oscillating multipliers on stratified groups Some sharp inequalities for norms in $$\mathbb {R}^n$$ and $$\mathbb {C}^n$$ Ill-posedness for the gCH-mCH equation in Besov spaces Stability of pseudo peakons for a new fifth order CH type equation with cubic nonlinearities