## Details

Original language | English |
---|---|

Pages (from-to) | 475-489 |

Number of pages | 15 |

Journal | Archiv der Mathematik |

Volume | 122 |

Issue number | 5 |

Early online date | 24 Mar 2024 |

Publication status | Published - May 2024 |

## 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.

## Keywords

- 05C25, 20E45, 20F16, Graph, Conjugacy class, Non-solvable group, Genus, Commuting probability

## ASJC Scopus subject areas

## Cite this

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**Genus and crosscap of solvable conjugacy class graphs of finite groups.**/ Bhowal, Parthajit; Cameron, Peter J.; Nath, Rajat Kanti et al.

In: Archiv der Mathematik, Vol. 122, No. 5, 05.2024, p. 475-489.

Research output: Contribution to journal › Article › Research › peer review

*Archiv der Mathematik*, vol. 122, no. 5, pp. 475-489. https://doi.org/10.1007/s00013-024-01974-2

*Archiv der Mathematik*,

*122*(5), 475-489. https://doi.org/10.1007/s00013-024-01974-2

}

TY - JOUR

T1 - Genus and crosscap of solvable conjugacy class graphs of finite groups

AU - Bhowal, Parthajit

AU - Cameron, Peter J.

AU - Nath, Rajat Kanti

AU - Sambale, Benjamin

PY - 2024/5

Y1 - 2024/5

N2 - 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 x∈C and y∈D such that ⟨x,y⟩ is solvable. In this paper, we discuss certain properties of the 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 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 Γsc(G) and the commuting probability of certain finite non-solvable group.

AB - 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 x∈C and y∈D such that ⟨x,y⟩ is solvable. In this paper, we discuss certain properties of the 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 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 Γsc(G) and the commuting probability of certain finite non-solvable group.

KW - 05C25, 20E45, 20F16

KW - Graph, Conjugacy class, Non-solvable group, Genus, Commuting probability

UR - http://www.scopus.com/inward/record.url?scp=85188421904&partnerID=8YFLogxK

U2 - 10.1007/s00013-024-01974-2

DO - 10.1007/s00013-024-01974-2

M3 - Article

AN - SCOPUS:85188421904

VL - 122

SP - 475

EP - 489

JO - Archiv der Mathematik

JF - Archiv der Mathematik

SN - 0003-889X

IS - 5

ER -