Verbally and existentially closed subgroups of free nilpotent groups

V. A. Roman'kov ; N. G. Khisamiev

Algebra i logika, Tome 52 (2013) no. 4, pp. 502-525

Résumé

Let $\mathcal N_c$ be the variety of all nilpotent groups of class at most $c$ and $N_{r,c}$ a free nilpotent group of finite rank $r$ and nilpotency class $c$. It is proved that a subgroup $H$ of $N_{r,c}$ ($r,c\ge1$) is verbally closed iff $H$ is a free factor (or, equivalently, an algebraically closed subgroup, a retract) of the group $N_{r,c}$. In addition, for $c\ge4$ and $m$, every free factor $N_{m,c}$ of the group $N_{c-1,c}$ in the variety $\mathcal N_c$ is not existentially closed in the group $N_{m+i,c}$ for $i=1,2,\dots$. It is stated that for $r\ge3$ and $2\le c\le3$ every free factor $N_{m,c}$, $2\le m\le r$, in $\mathcal N_c$ is existentially closed in the group $N_{r,c}$.

Détail

Keywords: verbally closed subgroup, existentially closed subgroup, free nilpotent group.
Mots-clés : retract