The normal quotient philosophy. for edge-transitive graphs. Cheryl E Praeger. University of Western Australia

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1 The normal quotient philosophy for edge-transitive graphs Cheryl E Praeger University of Western Australia 1

2 Edge-transitive graphs Graph Γ = (V, E): V = vertex set E = edge set { unordered pairs from V }. Arc of Γ: (u, v) such that {u, v} E Automorphism of Γ: edge-preserving permutation of V so automorphisms lie in Sym (V ) = { all permutations of V } = Symmetric group on V Automorphism group of Γ: Aut (Γ) Sym (V ) G Aut (Γ): G transitive on V G-vertex-transitive G transitive on E G-edge-transitive G transitive on arcs G-arc-transitive 2

3 G-edge-transitive graphs Γ Connectivity: Apart from possibly some isolated vertices, all connected components that contain edges are isomorphic So we assume that Γ is connected G-vertex-orbits: Either (1) G-vertex-transitive or (2) Γ bipartite and G-vertex orbits are the biparts 3

4 Scope of lecture origin: distance-transitive and s-arc transitive graphs normal quotients and quasiprimitive groups the philosophy : basic graphs vs typical graphs framework for understanding and analysis (locally) s-arc-transitive graphs evaluation and future directions 4

5 History I (1960s): arc-transitive case Two general construction methods: for finite arc-transitive graphs Each construction method produces up to isomorphism at least one copy of each finite arc-transitive graph Gert Sabidussi: Input: finite group G, subgroup H, element g G \ H, g 2 H. Output: arc-transitive graph, G acts arc-transitively. 5

6 History I (1960s): continued 1967 C. (Charlie) C. Sims: Input: transitive permutation group G Sym (V ) with G even. Output: at least one G-arc-transitive graph with vertex set V. an orbital graph: for vertex a and H = G a, a joined to all vertices in certain H-orbit Γ(a) = b H. General question: Graph properties Group Structure? 6

7 History II (1960/70s): Distance transitive graphs Γ = (V, E): for 0 i diameter, Γ i = {(u, v) distance(u, v) = i} Γ is G-distance transitive: G transitive on each Γ i Examples: Cycles C n, complete graphs K n, Odd graphs O n, Johnson graphs J(n, k) all distance transitive 7

8 Early work on distance transitive graphs D. G. Higman 1967: Permutation groups with maximal diameter Biggs, Smith, 1971: Valency 3. Let v V, H = G v. Suppose H < K < G; B := {K images of v}, P := corresponding partition V : only two possible kinds of partitions P. 8

9 Antipodal Partition and Bipartition Antipodal relation: u v u = v or d(u, v) = diameter Γ antipodal: if antipodal relation is an equivalence relation if so the antipodal partition is the set of equivalence classes Bipartition: possible iff Γ bipartite (Distance transitive Γ may be both antipodal and bipartite) Examples: Cubes Q n 9

10 Graph Quotients Graph Γ = (V, E): vertex partition P Quotient graph: Γ P = (P, E P ) where C, C P adjacent some u C, u C are adjacent in Γ. 10

11 Properties of Γ P for G-arc-transitive Γ Assume Γ is G-arc-transitive and connected and generate P as before: v V, H = G v, H < K < G, B := {K images of v}, P := set of G-images of B 1: Γ P is connected 2: B contains no edges of Γ 3: G also arc-transitive on Γ P (possibly with non-trivial kernel) P maximal G vertex-primitive and arc-transitive on Γ P 11

12 Γ d.t., antipodal, valency 3; antipodal partition P P > 2 G distance transitive on Γ P = (P, E P ) and Γ covers Γ P 12

13 Γ d.t., bipartite; bipartition P P = 2 Γ bipartite, P = {V 1, V 2 }, and Γ 1 = (V 1, E 1 ) distance transitive where {u, v} E 1 distance (u, v) = 2. D.H.Smith s Theorem

14 Finite primitive distance transitive graphs Each finite d.t.g. Γ: leads (quickly) to primitive d.t.g. (Cameron) 1979 O Nan Scott Theorem: divides finite primitive permutation groups into 8 disjoint classes. Two relevant types: Almost simple type: : T G Aut(T ), T nonabelian simple group. Affine type: : Z d p < G AGL (d, p), V = finite affine space 14

15 Finite primitive d.t.graphs almost classified Saxl, Yokoyama, CEP 1987: G vertex-primitive and d.t. on Γ Γ known or G of affine type or almost simple type. Prospective classification: Huge effort by many researchers. 15

16 Review the framework of this (almost) classification 1: Find reduction that links each graph in the family with a smaller one having additional nice properties and such that no further reduction possible. [Here vertex-primitive d.t. graph] 2: Call graphs in family with no possible reductions basic 3: Keep track of the possible links between typical graphs and their associated basic graphs. [Here bipartite doubles or antipodal covers.] 4: Classify the basic graphs in the family 5: Elucidate the structure of arbitrary graphs in the family in the light of knowing the basic examples 16

17 Vertex-transitive s-arc-transitive graphs s-arc: path of length s (s edges) possibly self-intersecting but consecutive edges must be different Examples: for Γ = C 5 (1, 0, 4) is a 2-arc (1, 2, 3, 4, 0, 1, 2) is a 6-arc but (2, 3, 2, 1) is not an s-arc

18 (G, s)-arc transitive graph Γ: G transitive on s-arcs of Γ Many famous and beautiful examples: Complete graphs K n (s = 2), complete bipartite graphs K n,n (s = 3), Odd graphs O n (s = 3 if n 3); and stunning sporadic examples, e.g. Cai Heng Li s Monster graph (s = 4) Problem with reduction route: if P is G-invariant partition, then Γ P not usually s-arc transitive far from it! 18

19 Answer: use only normal quotients Γ connected and (G, s)-arc transitive, s 2: 1 N G, N has 3 vertex-orbits P = P N : set of N-vertex-orbits (corresponds to G v < NG v < G) CEP (1985): G acts s-arc transitively on Γ PN with kernel N; and Γ covers Γ PN Γ PN : called a normal quotient of Γ 19

20 Reduction route for connected (G, s)-arc transitive Γ Choose: N G maximal such that P N 3. Consequence: Γ is cover of the (G/N, s)-arc transitive normal quotient Γ PN such that all nontrivial normal subgroups of G/N have at most two vertex orbits on Γ PN G/N quasiprimitive: every non-trivial normal subgroup transitive or G/N bi-quasiprimitive: not quasiprimitive, but every non-trivial normal subgroup has at most two orbits (here Γ, Γ PN both bipartite) 20

21 Thus in the family of finite s-arc transitive graphs each graph Γ a cover of a (bi)quasiprimitive s-arc transitive normal quotient basic (G, s)-arc transitive graphs have G (bi)quasiprimitive on vertices Contrast with Babai s 1985 result: indicated that s-arc transitive graphs form an untameable wild class of graphs 21

22 [Quasi]primitive permutation groups G Sym (Ω) = S n G quasiprimitive: every non-trivial normal subgroup transitive G primitive: G transitive and stabiliser G α < max G 1830 Evariste Galois* Galois primitive permutation groups were really quasiprimitive (P. M. Neumann 2005) * Second Mémoire, first published

23 Finite quasiprimitive permutation groups CEP 1993 Similar structure to O Nan Scott primitive types: Divided into several different types: affine (HA), almost simple (AS), diagonal (SD, CD), product action (PA), twisted wreath (TW),.... CEP 2006 Building blocks for finite transitive permutation groups: each finite transitive group G embeddable in both G 1 G 2 G r (iterated wreath product) each G i quasiprimitive and H 1 H 2 H r (iterated wreath product) each H i primitive Use primitive or quasiprimitive groups as appropriate to the application 23

24 Quasiprimitive s-arc transitive graphs CEP 1993: Γ (G, s)-arc transitive and G-vertex quasiprimitive (s 2) G is one of 4 of the possible 8 O Nan Scott types. affine type almost simple twisted wreath product action classified (Ivanov & CEP) classifications for some classes of small rank almost simple groups (Fang, Hassani, Nochefranca, Wang, CEP) good description (Baddeley) constructions (Li & Seress) 24

25 When is this framework/approach applicable? Locally Q graphs: all graphs Γ = (V, E) such that, for all v V, Γ 1 (v) (or action of G v on Γ 1 (v)) has property Q. Works well for families of arc-transitive graphs: locally quasiprimitive locally primitive, What about: Families of (bipartite) vertex-intransitive, edge-transitive graphs? Families of (half-arc transitive) vertex-transitive, edge-transitive but not arc-transitive graphs? 25

26 Locally (G, s)-arc transitive graphs Collaboration: Michael Giudici, Cai heng and CEP G vertex intransitive: two orbits 1, 2 ; N G 1. If N intransitive on both 1 and 2 then Γ PN is locally (G/N, s) arc transitive. Moreover, Γ is a cover of Γ PN. 2. If N transitive on 1 and intrans on 2 then Γ PN is a star. 26

27 Two types of basic locally (G, s)-arc trans. graphs (i) G acts faithfully and quasiprimitively on both 1 and 2. (ii) G acts faithfully on both 1 and 2 and quasiprimitively on only 1. (The star case) How it works: For general Γ, G, if N G is maximal and intransitive on both i, then Γ PN satisfies case (i) or case (ii), or Γ PN = K n,n 27

28 Outcomes for locally s-arc transitive case uses theory of quasiprimi- Substanial theory for basic examples: tive groups Unexpected constructions of new graphs: graphs admitting PSL(2, p n ) locally 5-arc transitive Unexpected new types of amalgams, reduction in problem of bounding s, etc: What about the half-arc transitive case? 28

29 (G, 2 1 )-arc transitive and (G, 1 2 )-locally primitive Examples include: (i) Γ = C 3, G = Z 3, (ii) valency 4, half-arc transitive graphs Possibilities for normal quotient Φ = Γ PN : (i) Γ bipartite, P N = bipartition, Φ = C 2 (call this case trivial) (ii) Φ = Cn trans case) (n 3) (equivalent of the star quotients for locally s-arc (iii) Γ covers Φ and Φ is (G/N, 1 2 )-arc transitive and (G/N, 1 2 )-locally primitive (good reduction!) 29

30 (iv) Φ is G-arc transitive, val(φ) = 1 2 val(γ), Γ is a 2-multicover of Φ would love some advice on this - induced subgraph between adjacent N-orbits is sc 2r and/but...