Genus and crosscap of solvable conjugacy class graphs of finite groups

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

*Corresponding author for this work

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Abstract

The solvable conjugacy class graph of a finite group G, denoted by Γsc(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 xC and yD such that ⟨x,y⟩ is solvable. In this paper, we discuss certain properties of genus and crosscap of Γsc(G) for the groups D2n, Q4n, Sn, An, and PSL(2,2d). 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 Γsc(G) in terms of order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of Γsc(G) and the commuting probability of certain finite non-solvable groups.
Original languageEnglish
Pages (from-to)475-489
JournalArchiv der Mathematik
Volume122
Issue number5
Early online date24 Mar 2024
DOIs
Publication statusE-pub ahead of print - 24 Mar 2024

Keywords

  • Graph
  • Conjugacy class
  • Non-solvable
  • Genus
  • Commuting probability

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