A nilpotency condition for finitely generated soluble groups

Delizia, Costantino

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 4, pp. 237-239

Résumé

We prove that if \( k > 1 \) is an integer and \( G \) is a finitely generated soluble group such that every infinite set of elements of \( G \) contains a pair which generates a nilpotent subgroup of class at most \( k \), then \( G \) is an extension of a finite group by a torsion-free \( k \)-Engel group. As a corollary, there exists an integer \( n \), depending only on \( k \) and the derived length of \( G \) , such that \( G / Z_{n} (G) \) is finite. For \( k 4 \), such \( n \) depends only on \( k \).