Endimioni, Gérard Finite groups satisfying the condition \(({\mathcal N},n)\). (Groupes finis satisfaisant la condition \(({\mathcal N},n)\).) (French) Zbl 0822.20023 C. R. Acad. Sci., Paris, Sér. I 319, No. 12, 1245-1247 (1994). A group \(G\) satisfies the condition \(({\mathcal N}, n)\) if any set of \(n + 1\) elements of \(G\) contains a pair which generates a nilpotent subgroup. If \(G\) is a finitely generated soluble group then the index of the hypercentre is bounded by a function of \(n\). By considering the minimal finite simple groups the author shows that if \(G\) is a finite group satisfying \(({\mathcal N}, n)\) and \(n \leq 20\) then \(G\) is soluble. It is noted that \(A_ 5\) satisfies \(({\mathcal N}, 21)\). Reviewer: M.J.Tomkinson (Glasgow) Cited in 4 Documents MSC: 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks 20E07 Subgroup theorems; subgroup growth 20D15 Finite nilpotent groups, \(p\)-groups Keywords:nilpotent subgroups; finitely generated soluble groups; hypercentre; minimal finite simple groups PDFBibTeX XMLCite \textit{G. Endimioni}, C. R. Acad. Sci., Paris, Sér. I 319, No. 12, 1245--1247 (1994; Zbl 0822.20023)