Abstract

The solvable conjugacy class graph of a finite group G, denoted by \(\Gamma _{sc}(G)\) , 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 \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle \) is solvable. In this paper, we discuss certain properties of the genus and crosscap of \(\Gamma _{sc}(G)\) for the groups \(D_{2n}\) , \(Q_{4n}\) , \(S_n\) , \(A_n\) , and \({{\,\mathrm{\mathop {\textrm{PSL}}}\,}}(2,2^d)\) . 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 \(\Gamma _{sc}(G)\) 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 \(\Gamma _{sc}(G)\) and the commuting probability of certain finite non-solvable group.

中文翻译：

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有限群可解共轭类图的亏格和交叉上限

摘要

有限群G的可解共轭类图，表示为\(\Gamma _{sc}(G)\) ，是一个简单的无向图，其顶点是G的非平凡共轭类和两个不同的共轭类C， 如果存在\(x \in C\)和\(y \in D\)使得\(\langle x, y\rangle \)可解，则D是相邻的。在本文中，我们讨论群\ (D_{2n}\) 、 \ (Q_{4n}\)、\( S_n\)、\(A_n\)和\({{\,\mathrm{\mathop {\textrm{PSL}}}\,}}(2,2^d)\)。特别是，我们确定所有正整数n，使得它们的可解共轭类图是平面、环形、双环形或三环形。我们还将根据某些群的中心阶数和共轭类数量获得\(\Gamma _{sc}(G)\)的下界。由此，我们将推导出\(\Gamma _{sc}(G)\)的属与某个有限不可解群的交换概率之间的关系。