On Intersection of Abelian and Minimal Nonabelian Subgroups in Finite Groups

Let G be a finite group with subgroups A and B. Denote by M = MG(A,B) (respectively, m = mG(A,B)) the set of all minimal by inclusion (respectively, by order) intersections of the form A ∩ Bg, where g ∈ G. Put minG(A,B) = hmi and MinG(A,B) = hMi. In 1994 we proved that if A and B are abelian subgroups, then MinG(A,B) 6 F(G). In the present paper, we give other proof of this result. Futhermore, we construct a finite group G such that it contan an abelian subgroup A, a minimal non-abelian subgroup B and elements g1 and g2 with A ∩ Bg1 6 F(G), A ∩ Bg2 66 F(G), |A ∩ Bg1 | = |A ∩ Bg2 | and A ∩ Bg1 , A∩Bg2 ∈ minG(A,B).We provide an example of a group G such that g1, g2 ∈ G A∩Bg1 , A∩Bg2 ∈ MinG(A,B), A∩Bg1 6 F(G), and A∩Bg2 66 F(G). Moreover, we show that there exists a group G with nilpotent subgroups A and B such that m ⊂ M and minG(A,B) < MinG(A,B).