We prove that if is an integer and is a finitely generated soluble group such that every infinite set of elements of contains a pair which generates a nilpotent subgroup of class at most , then is an extension of a finite group by a torsion-free -Engel group. As a corollary, there exists an integer , depending only on and the derived length of , such that is finite. For , such depends only on .
Given a field K of characteristic p > 2 and a finite group G, necessary and sufficient conditions for the unit group U(KG) of the group algebra KG to be centrally metabelian are obtained. It is observed that U(KG) is centrally metabelian if and only if KG is Lie centrally metabelian.
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In this paper we deal with the class of groups for which whenever we choose two infinite subsets , there exist two elements , such that . We prove that an infinite finitely generated soluble group in the class is in the class of -Engel groups. Furthermore, with , we show that if is infinite locally soluble or hyperabelian group then .