Verbally and existentially closed subgroups of free nilpotent groups

Author： Roman’kov V.   Khisamiev N.  

Publisher： Springer Publishing Company

ISSN： 0002-5232

Source： Algebra and Logic, Vol.52, Iss.4, 2013-09, pp. : 336-351

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Abstract

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 ≥ 1) 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 ≥ 4 and m < c− 1, 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,…. It is stated that for r ≥ 3 and 2 ≤ c ≤ 3, every free factor N m,c , 2 ≤ m ≤ r, in $ {{mathcal{N}}_c} $ is existentially closed in the group N r,c .

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