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It is proved that, in any finite group with nilpotent subgroups and and the condition for any in , is a subgroup of . This generalizes the author's theorem about intersections of Abelian subgroups in a finite group, since this holds, for example, for Hamiltonian subgroups and in . [ABSTRACT FROM AUTHOR]

We complete the proof of the theorem that any nilpotent subgroups , , and of a finite group satisfy the inclusion , where is the Fitting subgroup of and and are some elements in . When , we get an affirmative answer to Questions 17.40 and 19.37 from The Kourovka Notebook. The proof uses the classification of finite simple groups. [ABSTRACT FROM AUTHOR]

It is proved that for any nilpotent subgroups A and B in a finite group G with sporadic socle, there is an element g such that A ∩ Bg = 1. [ABSTRACT FROM AUTHOR]

Given a finite group G with socle isomorphic to L ( q), q ≥ 4, we describe, up to conjugacy, all pairs of nilpotent subgroups A and B of G such that A ∩ B ≠ 1 for all g ∈ G. [ABSTRACT FROM AUTHOR]