RUS  ENG
Full version
JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2023 Volume 20, Issue 2, Pages 854–858 (Mi semr1615)

This article is cited in 1 paper

Mathematical logic, algebra and number theory

One corollary of description of finite groups without elements of order $6$

A. S. Kondrat'evab, M. S. Nirovac

a N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS, S. Kovalevskaya St., 16, 620108, Yekaterinburg, Russia
b Ural Federal University, Ural Matematical Center, Mira St., 19, 620002, Yekaterinburg, Russia
c Kabardino-Balkarian State University named after H.M. Berbekov, Chernyshevsky St., 175, 360004, Nalchik, Russia

Abstract: Let $G$ be a finite group. The set of all prime divisors of the order of $G$ is denoted by $\pi(G)$. The Gruenberg-Kegel graph (the prime graph) $\Gamma(G)$ of $G$ is defined as the graph with the vertex set $\pi(G)$ in which two different vertices $p$ and $q$ are adjacent if and only if $G$ contains an element of order $pq$. If the order of $G$ is even, then $\pi_1(G)$ denotes the connected component of $\Gamma(G)$ containing $2$. It is actual the problem of describing finite groups with disconnected Gruenberg-Kegel graphs. In the present article, all finite non-solvable groups $G$ with $3 \in \pi(G)\setminus \pi_1(G)$ are determined.

Keywords: finite group, non-solvable group, Gruenberg-Kegel graph.

UDC: 512.54

MSC: 20D60, 05C25

Received July 25, 2023, published October 26, 2023

DOI: 10.33048/semi.2023.20.052



© Steklov Math. Inst. of RAS, 2024