On intersections of abelian and nilpotent subgroups in finite groups. I

Intersection of Solvable Hall subgroups in finite groups

“…by [17,Theorem 1]. So we assume that H is not abelian, so, by the proof of [15, Theorem 4], a Sylow r-subgroup of Ĝ is not abelian.…”

“…It is easy to see that H ∩ H x consists of diagonal matrices, so H ∩ H x is abelian. Therefore, by [17,Theorem 1], there exists y ∈ G such that…”

On Intersection of Primary Subgroups in the Group Aut(F4(2))

“…It give us a description of pairs of subgroups (A, B) with the condition min G (A, B) ̸ = 1 for primary subgroups and sometimes for nilpotent subgroups A and B. For example, in [2,Theorem 1] it is proved that Min G (A, B) F (G) for any pair of abelian subgroups A and B of G, where F (G) is the Fitting subgroup of G (the greatest normal nilpotent subgroup of G).…”

“…Denote by P {2,3} the parabolic subgroup which is generated by monomial elements n r2 , n r3 and unipotent subgroup U corresponding positive roots. The subgroup P {2,3} is invariant under graph automorphism τ of order 2, its Levy subgroups L is isomorphic to the Chevalley group B 2 (2). The product L⟨τ ⟩ is isomorphic to the group Aut(A 6 ).…”

On intersections of solvable Hall subgroups in finite simple exceptional groups of Lie type

“…[8] Let A be an abelian subgroup of a finite group G. Then there exists x ∈ G such that A ∩ A x ≤ F (G).…”