Groups whose prime graph on conjugacy class sizes has few complete vertices

JOURNAL OF ALGEBRA

卷:364

Article

Casolo, Carlo²  Dolfi, Silvio²  Pacifici, Emanuele¹  Sanus, Lucia³ 

[1] Univ Milan, Dipartimento Matemat F Enriques, I-20133 Milan, Italy

[2] Univ Florence, Dipartimento Matemat U Dini, I-50134 Florence, Italy

[3] Univ Valencia, Fac Matemat, Dept Algebra, E-46100 Valencia, Spain

关键词: Finite groups;    Conjugacy class sizes;    Prime graph;   

DOI  :  10.1016/j.jalgebra.2012.04.013

来源: Elsevier

【 摘 要 】

Let G be a finite group, and let Gamma(G) denote the prime graph built on the set of conjugacy class sizes of G. In this paper, we consider the situation when Gamma(G) has few complete vertices, and our aim is to investigate the influence of this property on the group structure of G. More precisely, assuming that there exists at most one vertex of Gamma(G) that is adjacent to all the other vertices, we show that G is solvable with Fitting height at most 3 (the bound being the best possible). Moreover, if Gamma(G) has no complete vertices, then G is a semidirect product of two abelian groups having coprime orders. Finally, we completely characterize the case when Gamma(G) is a regular graph. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free