On the intersections of solvable Hall subgroups in finite groups

Vdovin, E. P.; Zenkov, V. I.

doi:10.1134/s0081543809070219

Cited by 6 publications

(7 citation statements)

References 9 publications

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“…Theorem 1 reduces both parts of Problem 1 to the investigation of almost simple groups. Notice also that Theorem 1 generalizes the main result of [11] in the following way.…”

Section: Introductionsupporting

confidence: 62%

“…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. In the present paper we prove the following Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

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

Vdovin

2012

J. Algebra Appl.

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“…Theorem 1 reduces both parts of Problem 1 to the investigation of almost simple groups. Notice also that Theorem 1 generalizes the main result of [11] in the following way.…”

Section: Introductionsupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

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

Vdovin

2012

J. Algebra Appl.

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“…As noted in [4, Lemma 3], if b(G, H) 4 then G has at least 5 regular orbits on Ω 5 , so Theorem 1 establishes the desired condition for all almost simple sporadic groups (in a strong form), bringing us a step closer to a proof of Vdovin's conjecture. It also extends a special case of a theorem of Vdovin and Zenkov [33,Theorem 2], which states that b(G, H) 5 when G is an almost simple sporadic group and H is a soluble π-Hall subgroup of G.…”

Section: Introductionmentioning

confidence: 60%

On soluble subgroups of sporadic groups

Burness¹

2021

Preprint

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Let G be an almost simple sporadic group and let H be a soluble subgroup of G. In this paper we prove that there exists x, y ∈ G such that H ∩ H x ∩ H y = 1, which is equivalent to the bound b(G, H) 3 with respect to the base size of G on the set of cosets of H. This bound is best possible. In this setting, our main result establishes a strong form of a more general conjecture of Vdovin on the intersection of conjugate soluble subgroups of finite groups. The proof uses a combination of computational and probabilistic methods.

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“…In case (a), the proof of [57,Theorem 2] gives reg(L, 5) 12 and thus b(G, H) 5 as required. In fact, a straightforward Magma computation shows that reg(L, 5) = 600 in this case.…”

Section: Exceptional Groups: Non-parabolic Actionsmentioning

confidence: 95%

Base sizes for primitive groups with soluble stabilisers

Burness

2020

Preprint

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Let G be a finite primitive permutation group on a set Ω with point stabiliser H. Recall that a subset of Ω is a base for G if its pointwise stabiliser is trivial. We define the base size of G, denoted b(G, H), to be the minimal size of a base for G. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that b(G, H) 4 if G is soluble. In this paper we extend Seress' result by proving that b(G, H) 5 for all finite primitive groups G with a soluble point stabiliser H. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that 4 random elements in Ω form a base tends to 1 as |G| tends to infinity.

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On the intersections of solvable Hall subgroups in finite groups

Cited by 6 publications

References 9 publications

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

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

On soluble subgroups of sporadic groups

Base sizes for primitive groups with soluble stabilisers

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