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A comparison of the order components in Frobenius and 2-Frobenius groups with finite simple groups. (English) Zbl 1230.20013

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.

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
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