V.I. Murashka. Arithmetic graphs and factorized finite groups ... P. 181-192

The Hawkes graph $\Gamma_H(G)$ of a group $G$ is the directed graph with vertex set $\pi(G)$ that has an edge $(p, q)$ whenever $q\in\pi(G/O_{p',p}(G))$. The Sylow graph $\Gamma_s(G)$ of a group $G$ is the directed graph with vertex set $\pi(G)$ that has an edge $(p, q)$ whenever $q \in\pi(N_G(P)/PC_G(P))$ for some Sylow $p$-subgroup $P$ of $G$. The $N$-critical graph $\Gamma_{Nc}(G)$ of a group $G$ is the directed graph with vertex set $\pi(G)$ that has an edge $(p, q)$ whenever $G$ contains a Schmidt $(p, q)$-subgroup, i.e., a Schmidt $\{p, q\}$-subgroup with a normal Sylow $p$-subgroup. The paper studies the Hawkes, Sylow, and $N$-critical graphs of products of totally permutable, mutually permutable, and $\mathfrak{N}$-connected subgroups.

Keywords: finite group, Hawkes graph, Sylow graph, $N$-critical graph, product of totally permutable subgroups, product of mutually permutable subgroups, $\mathfrak{N}$-connected subgroups

Received June 9, 2023

Revised August 8, 2023

Accepted August 28, 2023

Funding Agency: This work was supported by the Belarusian Republican Foundation for Fundamental Research (project no. Φ23PHΦ-237).

Viachaslau Igaravich Murashka, Cand. Sci. (Phys.-Math.), Francisk Skorina Gomel State University, Gomel, 246028 Belarus, e-mail: mvimath@yandex.ru

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Cite this article as: V. I. Murashka. Arithmetic graphs and factorized finite groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 181–192.