Gruppi con identità semigruppali: su una congettura di M. V. Sapir

Longobardi, Patrizia ; Maj, Mercede ; Wiegold, James

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 3, pp. 191-196

Résumé

M. V. Sapir conjectured the following: let \( S \) be a finitely generated semigroup satisfying the identity \( x^{2} = 0 \), and assume that \( S \) is a homomorphic image of a subsemigroup of a nilpotent group \( G \). Then \( S \) is finite. If that is true, then a soluble group with a finite basis for its semigroup identities is either abelian or of finite exponent. In this paper we prove the conjecture if either \( \gamma_{3} (G) \) is periodic or \( S \) is \( 3 \)-generator and \( \gamma_{4} (G) \) is periodic.