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 -