Genus and crosscap of solvable conjugacy class graphs of finite groups

verfasst von

Parthajit Bhowal, Peter J. Cameron, Rajat Kanti Nath, Benjamin Sambale

Abstract

The solvable conjugacy class graph of a finite group G, denoted by Γ_sc(G), is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x∈C and y∈D such that ⟨x,y⟩ is solvable. In this paper, we discuss certain properties of the genus and crosscap of Γ_sc(G) for the groups D_2n, Q_4n, S_n, A_n, and PSL(2,2^d). In particular, we determine all positive integers n such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of Γ_sc(G) in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of Γ_sc(G) and the commuting probability of certain finite non-solvable group.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

Tezpur University
Cachar College
University of St. Andrews

Typ

Artikel

Journal

Archiv der Mathematik

Anzahl der Seiten

15

ISSN

0003-889X

Publikationsdatum

24.03.2024

Publikationsstatus

Elektronisch veröffentlicht (E-Pub)

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Mathematik (insg.)

Elektronische Version(en)

https://doi.org/10.1007/s00013-024-01974-2 (Zugang: Geschlossen)