The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

Swartz, Eric

doi:10.1016/j.jcta.2012.01.005

Cited by 7 publications

(8 citation statements)

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“…In our main theorem below, we also prove that the examples we construct are basic, with respect to the full automorphism group of the graph. This establishes that the graphs we construct are new, in the sense that they do not arise from one of the constructions in [8], or the sequels [10,11,12] or further studies of locally s-arc transitive graphs [7,9,13,15,24]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 78%

A new infinite family of star normal quotient graphs of twisted wreath type

Kaja¹,

Morgan²

2022

Preprint

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We construct the first infinite families of locally arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger's program for the classification of locally arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.

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Section: Introductionmentioning

confidence: 78%

A new infinite family of star normal quotient graphs of twisted wreath type

Kaja¹,

Morgan²

2022

Preprint

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“…Indeed, the nontrivial normal subgroups of any primitive group are transitive, and hence primitive groups are quasiprimitive. Locally s-arc-transitive graphs with a group of automorphisms acting quasiprimitively on at least one orbit of vertices have been studied extensively; see [11,12,15,16,17,24,32].…”

Section: Permutation Group Theory and Graph Symmetrymentioning

confidence: 99%

A classification of finite antiflag-transitive generalized quadrangles

Bamberg

Swartz

2017

Trans. Amer. Math. Soc.

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Ostrom and Wagner (1959) proved that if the automorphism group G of a finite projective plane π acts 2-transitively on the points of π, then π is isomorphic to the Desarguesian projective plane and G is isomorphic to PΓL(3, q) (for some prime-power q). In the more general case of a finite rank 2 irreducible spherical building, also known as a generalized polygon, the theorem of Fong and Seitz (1973) gave a classification of the Moufang examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group G acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to G being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.2010 Mathematics Subject Classification. Primary 51E12, 20B05, 20B15, 20B25. 1 2 JOHN BAMBERG, CAI HENG LI, AND ERIC SWARTZ Conjecture 1.1 (W. M. Kantor 1991). If Q is a finite flag-transitive generalized quadrangle and Q is not a classical generalized quadrangle, then (up to duality) Q is the unique generalized quadrangle of order (3, 5) or the generalized quadrangle of order (15, 17) arising from the Lunelli-Sce hyperoval.Finite generalized polygons satisfying stronger symmetry assumptions, such as the Moufang condition [15] or distance-transitivity [12,28], have been classified. The current state-of-the-art for generalized quadrangles is the classification of antiflag-transitive finite generalized quadrangles in [5], where it was shown that, up to duality, the only nonclassical antiflag-transitive generalized quadrangle is the unique generalized quadrangle of order (3,5). (An antiflag is a non-incident point-line pair.) Notice that by the GQ Axiom, antiflag-transitivity implies flag-transitivity for a generalized quadrangle.The aim of this paper is to provide further progress toward Conjecture 1.1. We study finite generalized quadrangles with a strictly weaker local symmetry condition, local 2transitivity. If G is a subgroup of collineations of a finite generalized quadrangle Q that is transitive both on pairs of collinear points and pairs of concurrent lines, then Q is said to be a locally (G, 2)-transitive generalized quadrangle.Our main result is a complete classification of thick locally 2-transitive generalized quadrangles, the proof of which relies on the Classification of Finite Simple Groups (CFSG) [20].Theorem 1.2. If Q is a thick finite locally (G, 2)-transitive generalized quadrangle and Q is not a classical generalized quadrangle, then (up to duality) Q is the unique generalized quadrangle of order (3, 5).An equivalent definition of a locally 2-transitive generalized quadrangle is that it has an incidence graph that is locally 2-arc-transitive; see Section 2 for details. Since an equivalent definition of an antiflag-transitive generalized quadrangl...

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“…It was shown in [17,Lemma 3.2] that all locally 2-arc-transitive graphs where the quasiprimitive action is of type HA on both orbits are actually vertex-transitive and so classified in [18]. All locally (G, 2)arc-transitive graphs have been classified in the cases where G is an almost simple group whose socle is a Ree group [7], Suzuki group [31], or PSL(2, q) [3], while the sporadic group case was studied in [20]. Examples also exist in the PA and TW cases as we can take standard double covers of s-arc-transitive graphs of type PA and TW respectively.…”

Section: Introductionmentioning

confidence: 99%

“…, 58, 47,10,15, 70, 62,13,32, 59, 57,31, 66,22,24, 67, 48,27,35, 50, 45,12,23,11, 52,4, 64, GAP, we see thatL ∼ = C 71 :C 70 × C 9 , R ∼ = C 19 :C 18 × C 35 , L ∩ R ∼ = C 630 , L, R = G,and by Lemma 3.1, the coset graph Cos(G, L, R) is a connected locally (G, 2)-arc-transitive graph.…”

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confidence: 99%

Locally s-arc-transitive graphs arising from product action

Giudici¹,

Swartz²

2022

Preprint

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We study locally s-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally (G, 2)-arc-transitive graph with G acting quasiprimitively with type PA on both G-orbits of vertices, the group G does not act primitively on either orbit. Moreover, we construct the first examples of locally s-arc-transitive graphs of PA type that are not standard double covers of s-arctransitive graphs of PA type, answering the existence question for these graphs.

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The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

Cited by 7 publications

References 11 publications

A new infinite family of star normal quotient graphs of twisted wreath type

A new infinite family of star normal quotient graphs of twisted wreath type

A classification of finite antiflag-transitive generalized quadrangles

Locally s-arc-transitive graphs arising from product action

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