On edge-primitive 3-arc-transitive graphs

Giudici, Michael; King, Carlisle S. H.

doi:10.1016/j.jctb.2021.06.012

Cited by 6 publications

(13 citation statements)

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“…We now consider N ∈ C 3 , so that N is of the form Cl m (q t ) for some classical group and prime t such that n = mt. By [15,Propositions 4.3.i,i=6,7,10,14,16,17], N ∩ T contains a normal subgroup of index t, and so t = 2. However, the same results imply that N ∩ T contains a classical group Cl n 2 (q 2 ), contradicting (3).…”

Section: Classical Groups: Faithful Local Actionmentioning

confidence: 99%

“…This fact can often be used to rule out certain pairs (E, H). (7) In a number of remaining cases (E, H) the valency equals 4, and we check the classification of tetravalent edge-primitive 2-arc-transitive graphs given in [10] to see if such a lattice exists. (8) This leaves the lattices listed in [17, Table 2] (which we add to our table) and one other possible lattice L = (J 1 , 7 : 6, 7 : 3, 2 3 : 7 : 3).…”

Section: Sporadic Groupsmentioning

confidence: 99%

“…(3) We cannot rule out any pairs by (3), as E is not equal to group number 1, 4 or 6. (4) For each (E, M) we consider the simple sections of M. For example, consider the pair (7,2). Here E = (A 4 × G 2 (4)) : 2 and M = 3.Suz : 2, and so…”

Section: Sporadic Groupsmentioning

confidence: 99%

“…However, this yields a contradiction using [5], as no maximal subgroup of X has order divisible by 2 14 .3 4 .5 2 .7.13. We therefore remove the pair (7,2). Using similar reasoning, we remove all but one pair.…”

Section: Sporadic Groupsmentioning

confidence: 99%

“…γδ PGL 2 (9) A 6 A 7 T q = p ≡ 23, 41 (mod 48) PΩ − 10 (7). γ, δ ′ Aut(A 6 ) S 6 S 7 M 22 .2, PSp 4 (7).2 [5] if G is sporadic. We conclude Section 3 with the proof of Theorem 1.…”

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confidence: 99%

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On edge-primitive 3-arc-transitive graphs

Giudici¹,

King²

2019

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This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.Edge-primitive graphs, that is, graphs whose automorphism group acts primitively on the set of edges, were first studied by Weiss in 1973 [32] who classified all the edge-primitive graphs of valency three. The study of edge-primitive graphs was reinvigorated in 2010 by the first author and Li [8] by providing a general structure theorem of such graphs and classifying all edge-primitive graphs whose automorphism group contains PSL 2 (q) as a normal subgroup. This has led to all edge-primitive graphs of valencies 4 [10] and 5 [11] being classified, and all those of prime valency and having a soluble edge-stabiliser [24]. Moreover, all edge-primitive graphs of prime power order [22] or which are Cayley graphs on abelian and dihedral groups [23] have been classified.An s-arc in a graph Γ is an (s + 1)-tuple (v 0 , v 1 , . . . , v s ) of vertices such that v i ∼ v i+1 but v i = v i+2 . We say that Γ is s-arc-transitive if the automorphism group of Γ acts transitively on the set of s-arcs. If all vertices of Γ have valency at least two then an s-arc-transitive graph is also (s − 1)-arc-transitive. The study of s-arc-transitive graphs originated in the seminal work of Tutte [30,31], who showed that a graph of valency three is at most 5-arc-transitive. This was later extended by Weiss [33] who showed that a graph of valency at least three is at most 7-arc-transitive. The vertex-primitive 4-arctransitive graphs were classified by Li [16] and all edge-primitive 4-arc-transitive graphs were classified by Li and Zhang [17]. These classifications were enabled by the classification of all vertex-stabiliser, edge-stabiliser pairs for 4-arc-transitive graphs by Weiss [33]. A key part of the latter classification is that the edge-stabiliser is always soluble. Han and Lu [12] have subsequently classified all edge-primitive graphs for almost simple groups with soluble edge-stabilisers.Edge-primitive 2-arc-transitive graphs have been investigated by Lu [20] who showed that if the graph Γ is not complete bipartite then the automorphism group is almost simple, that is, has a unique minimal normal subgroup T and T is a nonabelian simple group. He further showed that if Γ is 3-arc-transitive then either the graph has valency 7 and G v = A 7 or S 7 , or G v acts unfaithfully on the set Γ(v) of neighbours of v, where G = Aut(Γ). Note that for an edge-primitive graph, the edge-stabiliser G {u,v} is maximal in G. Moreover, the arc-stabiliser G u,v is an index two subgroup of G {u,v} and also contained in two other subgroups, namely the vertex-stabilisers G v and G w . These observations make

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