Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

Spiga, Pablo

doi:10.1515/jgt.2011.097

Cited by 17 publications

(11 citation statements)

References 13 publications

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“…The following counterexample is due to Pyber: let G = Sym(2n + 1) and A = H ∪ {σ, σ −1 }, where σ is the shift m → m+2 mod 2n+1 and H is the subgroup generated by all transpositions (i, i+1) with 1 ≤ i ≤ n. Then |A 3 | ≪ |A|. See also [PPSS12,§3] and [Spi12].…”

Section: Some Open Problemsmentioning

confidence: 99%

Growth in groups: ideas and perspectives

Helfgott

2015

Bull. Amer. Math. Soc.

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Section: Some Open Problemsmentioning

confidence: 99%

Growth in groups: ideas and perspectives

Helfgott

2015

Bull. Amer. Math. Soc.

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“…Assume for a contradiction that this is false. Then [35,Corollary 2] shows that there is a prime p such that G [1] xy and F * (G xy ) non-trivial are p-groups. If F * (G xy ) G [1] x , then we have that F * (G xy ) = G [1] x and so [12,…”

Section: Graph-theoretical Problemsmentioning

confidence: 99%

A theory of semiprimitive groups

Giudici

Morgan

2018

Journal of Algebra

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A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems arising in universal algebra related to collapsing monoids and the graphrestrictive problem for permutation groups. Here we develop a theory of semiprimitive groups which encompasses their structure, their quotient actions and a method by which all finite semiprimitive groups are constructed. We also extend some results from the theory of primitive groups to semiprimitive groups, and conclude with open problems of a similar nature. 1 primitive group are transitive. This leads to a natural generalisation where we call a permutation group quasiprimitive if each of its non-trivial normal subgroups is transitive. Many questions about permutation groups can be reduced to questions about primitive or quasiprimitive groups and they have been the focus of much attention, for example [1,8,9,10,13,14,16,17,19,23,25,26,27,28,30,31,32].Innately transitive permutation groups were introduced by Bamberg and Praeger [3] and these are the finite permutation groups G with a transitive minimal normal subgroup N. Such groups naturally occur as overgroups of quasiprimitive groups. A permutation group G is called semiregular if each point-stabiliser G ω is trivial. It is well known that the centraliser of a transitive group is semiregular [11, Theorem 4.2A] and so a normal subgroup of an innately transitive group either contains the transitive minimal normal subgroup N, and hence is itself transitive, or intersects N trivially and hence is semiregular.A permutation group is called semiprimitive if every normal subgroup is transitive or semiregular. This notion was introduced by Bereczky and Maróti [5] and was motivated by an application to collapsing transformation monoids. Their original definition required the group to be non-regular, but here we follow Potočnik, Spiga and Verret [22] and include the regular case. The class of semiprimitive groups is much wider than the class of innately transitive groups and includes all automorphism groups of graphs that are vertextransitive and locally quasiprimitive (see Lemma 8.1) and all finite Frobenius groups [5, Lemma 2.1]. Note that every normal subgroup of a semiregular group is also semiregular, therefore semiregular groups (whose theory is rather uninteresting) could be considered as the intransitive analogue of semiprimitive groups.Potočnik, Spiga and Verret were interested in semiprimitive groups due to their work on the Weiss Conjecture and its generalisations. A finite transitive permutation group L is called graph-restrictive if there is an absolute constant c(L) such that for any locally L graph-group pair (Γ, G), the order of a vertex stabiliser in G is at most c(L) (see Section 8 for more details). The Weiss Conjecture [38] asserts that any primitive group is graphrestrictive and has been proved for many classes of primitive groups, for example all 2transitive groups are graph-restrictive [36]. Praege...

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“…The construction used in the proof is a generalisation of the construction that appeared in [13,Section 4] (in fact, Theorem 1.3 generalises [13, Theorem 4]) which in turn was inspired by the so-called wreath extension construction [11,Chapter IV,8.1]. A precursory idea to this construction can also be traced to [19,Section 4]. Theorem 1.7 naturally leads one to wonder about the corresponding statement for amalgams of rank greater than two.…”

Section: Introductionmentioning

confidence: 99%

On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

2015

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Abstract. A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups L1 and L2 with L1 not semiprimitive, we construct an infinite family of rank two amalgams of permutation type [L1, L2] and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally [L1, L2] graphs.We also consider the corresponding question for amalgams of rank k ≥ 3. We completely resolve this by showing that the order of the Borel subgroup is bounded by the permutation type [L1, . . . , L k ] only in the trivial case where each of L1, . . . , L k is regular.

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Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

Cited by 17 publications

References 13 publications

Growth in groups: ideas and perspectives

Growth in groups: ideas and perspectives

A theory of semiprimitive groups

On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

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