University of Newcastle - Michael Giudici on joint work with ......Michael Giudici on joint work...

Local actions on graphs and semiprimitive groups

Michael Giudici

on joint work with Luke Morgan

Centre for the Mathematics of Symmetry and Computation

Winter of DisconnectednessNewcastle, July 2016

Automorphisms of graphs

Γ a locally finite, simple, connected graph.

Vertex set VΓ, edge set EΓ, arc set AΓ

Aut(Γ) is the group of all automorphisms of Γ.

Symmetry conditions

Given G 6 Aut(Γ) then G is

vertex-transitive: transitive on VΓedge-transitive: transitive on EΓarc-transitive: transitive on AΓ

Arc-transitive implies edge-transitive and vertex-transitive.

Edge-transitive but not vertex-transitive implies that Γ is bipartiteand G has two orbits on vertices.

Symmetry conditions

How many automorphisms?

G 6 Aut(Γ) arc-transitive or edge-transitive.

v ∈ VΓ, vertex stabiliser Gv .

Problem: Bound |Gv | when Gv finite?

• If Γ is finite and G is vertex-transitive, then by theOrbit-Stabiliser Theorem |G | = |VΓ||Gv |, so also bound |G |.

• If Γ is infinite then |Gv | is bounded if and only if Aut(Γ) hasfinitely many conjugacy classes of discrete arc/edge-transitivesubgroups.

Note that if Γ has valency d and G is arc-transitive then also have|Gv | = d |Gvw |.

A Theorem of Tutte(1947,1959)

Theorem Let Γ be a connected cubic graph with an arc-transitivegroup G of automorphisms such that Gv is finite. Then|Gv | = 3.2s for some s ≤ 4.

Structure of stabilisers

Djokovic and Miller (1980): Determined the possible structures offinite vertex and edge-stabilisers for cubic arc-transitive graphs:

• Only 7 possibilities for the pair (Gv ,Ge) with e = u, v.• In particular, G is a quotient of one of 7 finitely presented

groups.

Possibilities for (Gv ,Ge)

s Gv Ge

1 C3 C2

2 S3 C2 × C2 or C4

3 S3 × C2 D8

4 S4 D16 or QD16

5 S4 × C2 (D8 × C2) o C2

Applications

Conder and Dobcsanyi (2002): Determined all cubic arc-transitivegraphs on at most 768 vertices:

• |Aut(Γ)| 6 768.48 = 36864

• So need to find all normal subgroups of index at most 36864.

• Conder has subsequently enumerated all such graphs on atmost 10,000 vertices.

Applications

Conder and Dobcsanyi (2002): Determined all cubic arc-transitivegraphs on at most 768 vertices:

• |Aut(Γ)| 6 768.48 = 36864

• So need to find all normal subgroups of index at most 36864.

• Conder has subsequently enumerated all such graphs on atmost 10,000 vertices.

Edge-transitive

Goldschmidt (1980): Determined the possible structures of finitepairs (Gu,Gv ) for adjacent vertices u, v in cubic edge-transitivegraphs:

• only fifteen possibilities

• |Gv | 6 384

Local actions

Γ(v) is the set of neighbours of v .

Γ (v)

GΓ(v)v is the permutation group induced on Γ(v) by Gv , called the

local action of Gv .

If G is vertex-transitive then all the GΓ(v)v are isomorphic.

Local actions

Γ connected, G 6 Aut(Γ) vertex-transitive

• Given a permutation group L, we say that the pair (Γ,G ) is

locally L if GΓ(v)v∼= L for all vertices v .

• Given some permutation group property P, we say that (Γ,G )

is locally P if GΓ(v)v has property P for all vertices v .

Weiss Conjecture

Let G 6 Sym(Ω).

Call G primitive if the only partitions of Ω that it preserves are thetrivial ones Ω and ω | ω ∈ Ω.

Weiss Conjecture (1978): There is some function f (d) such thatfor every locally primitive pair (Γ,G ) of valency d and Gv finite wehave |Gv | 6 f (d).

• Tutte’s result is that f (3) = 48.

Weiss Conjecture

Let G 6 Sym(Ω).

Call G primitive if the only partitions of Ω that it preserves are thetrivial ones Ω and ω | ω ∈ Ω.

Weiss Conjecture (1978): There is some function f (d) such thatfor every locally primitive pair (Γ,G ) of valency d and Gv finite wehave |Gv | 6 f (d).

• Tutte’s result is that f (3) = 48.

Graph-restrictive

Verret: We say that L is graph-restrictive if there is a constant Csuch that for all locally L pairs (Γ,G ) with Gv finite, we have that|Gv | 6 C .

• Tutte’s result is that C3 and S3 are graph-restrictive.

• The Weiss Conjecture asserts that every primitive group isgraph-restrictive.

Graph-restrictive

A nonexampleWreath graphs

Aut(Γ) = S2 wrD2n

Aut(Γ)Γ(v)v = D8

|Aut(Γ)v | = 2n−1.2

An equivalent definition

G[i ]v is the kernel of the action of Gv on the set of all vertices at

distance at most i from v .

G[1]vw is the kernel of the action of Gvw on Γ(v) ∪ Γ(w), wherev ,w is an edge.

Lemma If Γ is connected and G[i ]v = G

[i+1]v for some i , then

G[i ]v = 1.

Lemma L is graph-restrictive if and only if there is some constant ksuch that for all locally L pairs (Γ,G ) with Gv finite, we have

G[k]v = 1.

Tutte: For cubic arc-transitive graphs with Gv finite we have

G[3]v = 1.

G[i ]v = 1.

G[k]v = 1.

G[3]v = 1.

G[i ]v = 1.

G[k]v = 1.

G[3]v = 1.

Some graph-restrictive groups

• Any regular group. (regular=free+transitive)

• Gardiner (1973): Any transitive subgroup of S4 other than D8.

• Sami (2006): D2n for n odd.

• Potocnik, Spiga, Verret (2012): GL(2, p) acting on the set ofnonzero vectors of GF(p)2.

Progress on the Weiss Conjecture

Trofimov, Weiss (1995): Any 2-transitive group is graph-restrictive

(G[6]v = 1)

Weiss (1979), Spiga (2016): If L is a primitive permutation group

of affine type, then L is graph-restrictive. (G[4]v = 1)

Spiga (2011): If L is a primitive permutation group with a regularnonabelian minimal normal subgroup then L is graph-restrictive.

(G[1]uv = 1)

Trofimov, Weiss (1995): PSLn(q) acting on m-spaces isgraph-restrictive.

(G[6]v = 1)

(G[1]uv = 1)

(G[6]v = 1)

(G[1]uv = 1)

(G[6]v = 1)

(G[1]uv = 1)

What is the correct setting?

Let G 6 Sym(Ω).

• Call G quasiprimitive if every nontrivial normal subgroup istransitive.

• Call G semiprimitive if every nontrivial normal subgroup istransitive or semiregular.(A permutation group H is semiregular on Ω if Hα = 1 for allα ∈ Ω, that is, free.)

Let G 6 Sym(Ω).

• Call G quasiprimitive if every nontrivial normal subgroup istransitive.

• Call G semiprimitive if every nontrivial normal subgroup istransitive or semiregular.(A permutation group H is semiregular on Ω if Hα = 1 for allα ∈ Ω, that is, free.)

Semiprimitive groups

Initially studied by Bereczky and Maroti (2008) (motivated by anapplication from universal algebra and collapsing monoids).

Examples include:

• primitive and quasiprimitive groups;

• regular groups;

• Frobenius groups (that is, all nontrivial elements fix at mostone point);

• GL(n, p) acting on the set of nonzero vectors of Znp.

• Any locally quasiprimitive, vertex-transitive group ofautomorphisms of a non-bipartite graph. (Praeger 1985)

Praeger Conjecture (2000): Every quasiprimitive group isgraph-restrictive.

Potocnik, Spiga, Verret (2012): If a transitive group is graphrestrictive then it is semiprimitive.

D8 is not semiprimitive as it contains a normal intransitivesubgroup isomorphic to C 2

2 that is not semiregular.

PSV conjecture: A transitive group is graph-restrictive if and onlyif it is semiprimitive.

The edge-transitive case

Γ edge-transitive but not vertex transitive. Edge v ,w

Say (Γ,G ) is locally [L1, L2] if GΓ(v)v∼= L1 or L2 for all vertices v .

Goldschmidt-Sims Conjecture: If L1 and L2 are primitive then thereis a constant C such that if (Γ,G ) is locally [L1, L2] with finitevertex stabilisers then |Gvw | ≤ C .

Morgan, Spiga, Verret (2015): If either L1 or L2 is notsemiprimitive then there is no bound on |Gvw | for a locally [L1, L2]pair (Γ,G ) with finite stabilisers.

Variation on Thompson-Wielandt

Given an edge v ,w, G [1]vw is the kernel of the action of Gvw on

Γ(v) ∪ Γ(w).

Thompson-Wielandt Theorem: If (Γ,G ) is a locally primitive pair

with Gv finite and v ,w is an edge, then G[1]vw is a p-group for

some prime p.

• van Bon (2003): Still holds if (Γ,G ) is locally quasiprimitive.

• Spiga (2012): Still holds if (Γ,G ) is locally semiprimitive.

Variation on Thompson-Wielandt

Given an edge v ,w, G [1]vw is the kernel of the action of Gvw on

Γ(v) ∪ Γ(w).

Thompson-Wielandt Theorem: If (Γ,G ) is a locally primitive pair

with Gv finite and v ,w is an edge, then G[1]vw is a p-group for

some prime p.

• van Bon (2003): Still holds if (Γ,G ) is locally quasiprimitive.

• Spiga (2012): Still holds if (Γ,G ) is locally semiprimitive.

Plinths

G 6 Sym(Ω), transitive.

Define a plinth of G to be a minimal transitive normal subgroupof G .

• Every finite transitive group has a plinth.

• If a group has a transitive minimal normal subgroup it is aplinth.

• Any regular normal subgroup is a plinth.

Properties of plinths of primitive groups

G primitive with minimal normal subgroup (plinth) N:

• N is characteristically simple and so in finite case N ∼= T k , forsome finite simple group T .

• CG (N) is semiregular.

• G has at most two plinths

• If M is a second plinth then N ∼= M and both N and M areregular.

O’Nan–Scott Theorem for primitive groups, quasiprimitive groups.

Properties of plinths of primitive groups

G primitive with minimal normal subgroup (plinth) N:

• N is characteristically simple and so in finite case N ∼= T k , forsome finite simple group T .

• CG (N) is semiregular.

• G has at most two plinths

• If M is a second plinth then N ∼= M and both N and M areregular.

O’Nan–Scott Theorem for primitive groups, quasiprimitive groups.

Structure of plinths of semiprimitive groups

A semiprimitive group can have arbitrarily many plinths, they maynot be isomorphic, and need not be characteristically simple.

Theorem (G-Morgan) Let K be a plinth of a semiprimitive group.

• If K is a nonregular plinth then K is perfect and unique.

• If L is a another plinth then every plinth is contained in KLand every nonregular normal subgroup contains KL.

Bereczky, Maroti (2008): A finite soluble semiprimitive group has aunique plinth, it is regular, and contains every intransitive normalsubgroup.

“Topological” plinths

Γ an infinite, locally finite, nonbipartite graph.

Let G be a non-discrete, vertex-transitive, locally quasiprimitiveclosed subgroup of Aut(Γ). Note that G is semiprimitive.

Define G (∞) = ∩L<GL, for L open and of finite index.

Burger-Mozes (2000): Let N be a closed normal subgroup of G .Then either:

• N is nondiscrete and contains the transitive group G (∞), or

• N is discrete and acts freely with infinitely many orbits.

Moreover, G (∞) is topologically perfect.

Multiple plinths

Theorem (G-Morgan) Let G be semiprimitive with distinct plinthsK and L. Then

• G = G/(K ∩ L) acts primitively on the set of (K ∩ L)-orbits

• there exists a characteristically simple group X such thatL/(K ∩ L) ∼= K/(K ∩ L) ∼= X .

Corollary Finite plinths have the same set of composition factors.

Theorem (G-Morgan) If G is a finite semiprimitive group with

multiple plinths then G is graph-restrictive. (G[1]uv = 1)

Multiple plinths

Nilpotent plinths

Let L be a finite semiprimitive group with a nilpotent plinth K .

Theorem (G-Morgan (2015)) Let (Γ,G ) be a locally L pair with Gv

finite and valency coprime to 6. Then L is graph-restrictive.

(G[1]vw = 1)

• Also give detailed information about what a counterexamplewith valency not coprime to 6 must look like.

• Analogous to Weiss’s results for primitive affine groups

University of Newcastle - Michael Giudici on joint work with ......Michael Giudici on joint work...

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