On finite 4-primary groups having a disconnected Gruenberg-Kegel graph and a composition factor isomorphic to $L_3(17)$ or $Sp_4(4)$

A. S. Kondrat'ev ; I. D. Suprunenko ; I. V. Khramtsov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 1, pp. 139-155

Résumé

The Gruenberg–Kegel graph (the prime graph) $\Gamma(G)$ of a finite group $G$ is the graph in which the vertices are the prime divisors of the order of $G$ and two distinct vertices $p$ and $q$ are adjacent if and only if $G$ contains an element of order $pq$. Investigations of finite groups by the properties of their Gruenberg–Kegel graphs form a dynamically developing branch of the finite group theory. A detailed study of the class of finite groups with disconnected Gruenberg–Kegel graphs is one of the important problems in this direction. In 2010–2011, the first and the third authors described the normal structure of finite 3-primary and 4-primary groups with disconnected Gruenberg–Kegel graphs. Unfortunately, the case where a 4-primary group has a composition factor isomorphic to $L_3(17)$ or $Sp_4(4)$ has been omitted in this description. In the present paper, we obtain a description of the groups under consideration in the omitted case. Now a description of the normal structure of finite 4-primary groups with disconnected Gruenberg–Kegel graphs is corrected. In the course of the proof, the 2-modular decomposition matrix of the group $L_3(17)$ is calculated (up to two parameters every of which takes value 1 or 2).

Détail

Keywords: finite group, non-solvable $4$-primary group, chief factor, disconnected Gruenberg–Kegel graph, character, Brauer character
Mots-clés : algebraic group, decomposition matrix.