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

Band

122

Seiten

475-489

Anzahl der Seiten

15

ISSN

0003-889X

Publikationsdatum

05.2024

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Mathematik (insg.)

Elektronische Version(en)

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