On intersections of solvable Hall subgroups in finite nonsolvable groups

On the Base Size of a Transitive Group With Solvable Point Stabilizer

“…It is also easy to see, that for every solvable subgroup T of Sym 5 = Aut(Alt 5 ) we have Base T (Sym 5 ) 4. In this case we have Reg Sym 4 (Sym 5 , 4) = 1 and the lemma from [12] implies that Base S (G) = 5.…”

“…First we show that the condition Reg S (Aut G (G i , G i−1 ), k) 5 is essential. The following example is given by V.I.Zenkov in [12].…”

“…In [4] S.Dolfi proved that in every π-solvable group G there exist elements x, y ∈ G such that the equality H ∩ H x ∩ H y = O π (G) holds, where H is a π-Hall subgroup of G (see also [10]). V.I.Zenkov in [12] constructed an example of a group G with a solvable π-Hall subgroup H such that the intersection of five subgroups conjugate with H in G is equal to O π (G), while the intersection of every four conjugates of H is greater than O π (G) (see Example 9 below). In [11] it is proven that if, for every almost simple group S possessing a solvable π-Hall subgroup H, the inequalities Base H (S ) 5 and Reg H (S , 5) 5 hold, then for every group G possessing a solvable π-Hall subgroup H the inequality Base H (G) 5 holds.…”

Intersection of Solvable Hall subgroups in finite groups

“…In 1966 D.S.Passman proved (see [10]) that a p-solvable group possesses three Sylow p-subgroups whose intersection equals the p-radical of G. Later in 1996 V.I.Zenkov proved (see [18]) that the same conclusion holds for arbitrary finite group G. In [4] S.Dolfi proved that in every π-solvable group G there exist three conjugate π-Hall subgroups whose intersection equals O π (G) (see also [13]). Notice also that V.I.Zenkov in [19] constructed an example of a group G possessing a solvable π-Hall subgroup H such that the intersection of five conjugates of H equals O π (G), while the intersection of every four conjugates of H is greater than O π (G).…”

“…In [19] It was conjuctured that if H is a solvable Hall π-subgroup of a finite group G, then Base H (G) ≤ 5. The following theorem allows to reduce the conjecture to the case of almost simple groups.…”