Infinite locally soluble \( k \)-Engel groups

Spiezia, Lucia Serena

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 3 (1992) no. 3, pp. 177-183

Résumé

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