On universal nilpotent groups (Q1120674)

A group G in a class \({\mathcal K}\) of groups is called universal in \({\mathcal K}\) if any group H in \({\mathcal K}\) of power at most that of G is embeddable in G. The paper investigates the existence of countable universal groups in various classes of nilpotent groups. Earlier work of \textit{D. Saracino} and \textit{C. Wood} [J. Algebra 58, 189-207 (1979; Zbl 0673.03024)] and the author [Ann. Pure Appl. Logic 35, 205-246 (1987; Zbl 0634.03027)] on existentially closed groups demonstrates the existence of countable universal groups in three classes of nilpotent groups: all nilpotent groups of class 2 and finite exponent dividing n; all locally finite nilpotent groups of class 2; all torsion free nilpotent groups of class at most c. (Here n and c are natural numbers.) In this paper some other classes are considered. It is shown that if \({\mathcal K}\) is either the class of nilpotent groups of class at most c (c\(\geq 2)\) or the class of nilpotent groups of class at most c for which the torsion subgroup is a p-group (p is prime) then \({\mathcal K}\) does not have a countable universal member. If we take \({\mathcal K}\) to be the class of nilpotent groups of class c and with torsion subgroup of exponent dividing n then the situation is not so clear. It is shown that, if all prime divisors of n are greater than c then an existentially closed group in \({\mathcal K}\) is the direct product of existentially closed groups in the two subclasses of groups of exponent dividing n and torsion free groups in \({\mathcal K}\). As a consequence it can be deduced that, in the case that \(c=2\), \({\mathcal K}\) contains countable universal groups.