Absolutely closed groups in the class of 2-step nilpotent torsion-free groups. (Q745641)

The dominion of a subgroup \(H\) in a group \(G\) (in a class \(M\)) is the set of all elements \(a\in G\) such that for every homomorphisms \(f,g\colon G\to B\in M\) if \(f,g\) coincide on \(H\) then \(a^f=a^g\). A group \(H\in M\) is \(n\)-closed in \(M\) if the dominion of \(H\) coincides with \(H\) for every group \(G\) in \(M\) containing \(H\) and generated modulo \(H\) by \(n\) appropriate elements. \(H\) is absolutely closed in \(M\) if \(H\) is \(n\)-closed in \(M\) for each \(n\). It is proved that divisible groups and only these groups are absolutely closed in the class of 2-nilpotent torsion-free groups. It is established that the additive group of the rationals is 1-closed in an arbitrary quasivariety of nilpotent torsion-free groups and 3-closed in an arbitrary quasivariety of 2-nilpotent torsion-free groups.