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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2022 Volume 63, Number 5, Pages 1010–1026 (Mi smj7709)

This article is cited in 1 paper

On the Hawkes graphs of finite groups

A. F. Vasil'ev, V. I. Murashka, A. K. Furs

Francisk Skaryna Gomel State University, Faculty of Mathematics

Abstract: We study the properties and applications of the directed graph, introduced by Hawkes in 1968, of a finite group $G$. The vertex set of $\Gamma_H(G)$ coincides with $\pi(G)$ and $(p,q)$ is an edge if and only if $q\in \pi(G/O_{p',p}(G))$. In the language of properties of this graph we obtain commutation conditions for all $p$-elements with all $r$-elements of $G$, where $p$ and $r$ are distinct primes. We estimate the nilpotence length of a solvable finite group in terms of subgraphs of its Hawkes graph. Given an integer $n > 1$, we find conditions for reconstructing the Hawkes graph of a finite group $G$ from the Hawkes graphs of its $n$ pairwise nonconjugate maximal subgroups. Using these results, we obtain some new tests for the membership of a solvable finite group in the well-known saturated formations.

Keywords: finite group, maximal subgroup, directed graph, Hawkes graph, arithmetic length of a solvable group, $C$-equivalent maximal subgroups, crown-equivalent maximal subgroups, hereditary saturated formation.

UDC: 512.542

MSC: 35R30

Received: 05.02.2022
Revised: 08.06.2022
Accepted: 15.06.2022

DOI: 10.33048/smzh.2022.63.504


 English version:
Siberian Mathematical Journal, 2022, 63:5, 849–861


© Steklov Math. Inst. of RAS, 2025