Regular orbits of symmetric and alternating groups

Fawcett, Joanna B.; O’Brien, E. A.; Saxl, Jan

doi:10.1016/j.jalgebra.2016.02.018

Cited by 18 publications

(22 citation statements)

References 30 publications

(171 reference statements)

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“…We begin by handling the cases where H 0 is an alternating or sporadic group. This is an easy application of the results of Fawcett et al [11,12] [15, p.185] for the definition), or H has a regular orbit on V . As explained in Section 1, if H has a regular orbit then G is not extremely primitive.…”

Section: Proof Of Theoremmentioning

confidence: 92%

A note on extremely primitive affine groups

Burness

Thomas

2020

Arch. Math.

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Let G be a finite primitive permutation group on a set $$\Omega $$ Ω with non-trivial point stabilizer $$G_{\alpha }$$ G α . We say that G is extremely primitive if $$G_{\alpha }$$ G α acts primitively on each of its orbits in $$\Omega {\setminus } \{\alpha \}$$ Ω \ { α } . In earlier work, Mann, Praeger, and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall’s conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.

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Section: Proof Of Theoremmentioning

confidence: 92%

A note on extremely primitive affine groups

Burness

Thomas

2020

Arch. Math.

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“…In this setting, the base-two problem can be viewed as a natural problem in representation theory (indeed, it is closely related to the famous k(GV )-problem [43], which establishes part of a conjecture of Brauer on defect groups of blocks). For example, see [23,24] for some recent results on base-two affine groups with a quasisimple point stabiliser.…”

Section: Introductionmentioning

confidence: 99%

On the Saxl graph of a permutation group

Burness¹,

Giudici²

2018

Math. Proc. Camb. Phil. Soc.

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“…Jan Saxl then initiated an ambitious project to classify the primitive permutation groups with base size 2. Much progress has been made on this problem for primitive groups of almost simple type [11,12,14,35,36,47], diagonal type [24] and affine type [26,27,31,[37][38][39]. Our first main result establishes that a large class of primitive twisted wreath products have base size 2.…”

Section: Introductionmentioning

confidence: 90%

Bases of twisted wreath products

Fawcett

2021

Preprint

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We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product G = T k :P acting on its base group Ω = T k , where T is some non-abelian simple group and P is some group acting transitively on k = {1, . . . , k} with k 2. We prove that if G is primitive on Ω and P is quasiprimitive on k, then G has base size 2. We also prove that the proportion of pairs of points that are bases for G tends to 1 as |G| → ∞ when G is primitive on Ω and P is primitive on k. Lastly, we determine the base size of any quasiprimitive group of twisted wreath type up to four possible values (and three in the primitive case). In particular, we demonstrate that there are many families of primitive groups of twisted wreath type with arbitrarily large base sizes.

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Regular orbits of symmetric and alternating groups

Cited by 18 publications

References 30 publications

A note on extremely primitive affine groups

A note on extremely primitive affine groups

On the Saxl graph of a permutation group

Bases of twisted wreath products

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