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A nilpotency condition for finitely generated soluble groups

Costantino Delizia (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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 .

Group algebras with centrally metabelian unit groups.

Meena Sahai (1996)

Publicacions Matemàtiques

Given a field K of characteristic p &gt; 2 and a finite group G, necessary and sufficient conditions for the unit group U(KG) of the group algebra KG to be centrally metabelian are obtained. It is observed that U(KG) is centrally metabelian if and only if KG is Lie centrally metabelian.

Groups generated by two mutually Engel periodic elements

H. Heineken (2000)

Bollettino dell'Unione Matematica Italiana

Scriviamo [ x , y ] = [ x , 1 y ] ed [ [ x , k y ] , y ] = [ x , k + 1 y ] . Cerchiamo gruppi S ⁢ L ⁢ 2 , q con generatori x , y tali che [ x , m y ] = x ed [ y , n x ] = y per alcuni numeri naturali m , n .

Infinite locally soluble k -Engel groups

Lucia Serena Spiezia (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we deal with the class E k * of groups G for which whenever we choose two infinite subsets X , Y there exist two elements x ∈ X , y ∈ Y such that x , y , … , y ︸ k = 1 . We prove that an infinite finitely generated soluble group in the class E k * is in the class E k of k -Engel groups. Furthermore, with k = 2 , we show that if G ∈ E 2 * is infinite locally soluble or hyperabelian group then G ∈ E 2 .

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