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Finite groups satisfying the condition \(({\mathcal N},n)\). (Groupes finis satisfaisant la condition \(({\mathcal N},n)\).) (French) Zbl 0822.20023

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 \(({\mathcal N}, 21)\).

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
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