Genus and crosscap of solvable conjugacy class graphs of finite groups

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 x∈C and y∈D 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.