EXISTENCE OF POLYNILPOTENT COVERING GROUPS OF A POLYNILPOTENT GROUP

Let Nc1, …., ct be the variety of polynilpotent groups of class row (c1, …, ct) It this paper, first, we show that a polynilpotent group G of class row (c1, …, ct) has no any Nc1, …, ct, ct+1 -covering group if this Baer-invariant with pespect to the variety Nc1, ..., ct,ct+1 is nontrivial. As an immediate consequence, we can conclude that a solvable group G of length c with nontrivial solvable multiplier, SnM (G), has no Sn-convering group for all n>c, where Sn is the variety of solvable groups of length at most n. Second, we prove that if G is a polynilpotent group of class row (c1, ..., ct,ct+1) such that Nc′1, ..., c′t, c′t+1 M(G) ≠1, where c′i ≥ci for all 1≤ i ≤ t and c′t+1 > ct+1, then G has no any Nc′1, ..., c′t, c′t+1 -covering group.

EXISTENCE OF POLYNILPOTENT COVERING GROUPS OF A POLYNILPOTENT GROUP Keywords:

V- convernig group , polynilpotent groups , solvable- group