On absolutely-nilpotent of class \( k \) groups

Longobardi, Patrizia ; MacHenry, Trueman ; Maj, Mercede ; Wiegold, James

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 6 (1995) no. 4, pp. 201-209

Résumé

A group \( G \) in a variety \( \mathfrak{V} \) is said to be absolutely-\( \mathfrak{V} \), and we write \( G \in A \mathfrak{V} \), if central extensions by \( G \) are again in \( \mathfrak{V} \). Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class \( A \mathfrak{N}_{k} \) of absolutely-nilpotent of class \( k \) groups. We prove some closure properties of the class \( A \mathfrak{N}_{k} \) and we show that every nilpotent of class \( k \) group can be embedded in an \( A \mathfrak{N}_{k} \)-gvoup. We describe all metacyclic \( A \mathfrak{N}_{k} \)-groups and we characterize \( 2 \)-generator and infinite \( 3 \)-generator \( A \mathfrak{N}_{2} \)-groups. Finally we study extensions \( 1 \to N \to H \to G \to 1 \), with \( N \le \zeta_{n} (H) \), the \( n \)-centre of \( H \), with \( n > 1 \).