Moghaddamfar, A. R. A comparison of the order components in Frobenius and 2-Frobenius groups with finite simple groups. (English) Zbl 1230.20013 Taiwanese J. Math. 13, No. 1, 67-89 (2009). Summary: Let \(G\) be a finite group. Based on the Gruenberg-Kegel graph \(\text{GK}(G)\), the order of \(G\) can be divided into a product of coprime positive integers. These integers are called the order components of \(G\) and the set of order components is denoted by \(\text{OC}(G)\). In this article we prove that if \(S\) is a non-Abelian finite simple group with a disconnected graph \(\text{GK}(S)\), with the exception of \(U_4(2)\) and \(U_5(2)\), and \(G\) is a finite group with \(\text{OC}(G)=\text{OC}(S)\), then \(G\) is neither Frobenius nor 2-Frobenius. For a group \(S\) isomorphic to \(U_4(2)\) or \(U_5(2)\), we construct examples of 2-Frobenius groups \(G\) such that \(\text{OC}(S)=\text{OC}(G)\). In particular, the simple groups \(U_4(2)\) and \(U_5(2)\) are not recognizable by their order components. Cited in 1 Document MSC: 20D05 Finite simple groups and their classification 20D60 Arithmetic and combinatorial problems involving abstract finite groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20D06 Simple groups: alternating groups and groups of Lie type Keywords:Frobenius groups; 2-Frobenius groups; Gruenberg-Kegel graphs; order components; finite groups; prime graphs; sets of element orders; finite simple groups; recognizable groups PDFBibTeX XMLCite \textit{A. R. Moghaddamfar}, Taiwanese J. Math. 13, No. 1, 67--89 (2009; Zbl 1230.20013) Full Text: DOI