Fixed point free elements of prime order - School of Maths and Stats ...

Fixed point free elements of prime orderMichael GiudiciUniversity of Western Australia

Orbit-Counting LemmaFor a finite group G acting on a set Ω,# of orbits = 1 ∑| fix(g)||G|g∈G

How many?Let n = |Ω| > 1,G a transitive subgroup of Sym(Ω).δ(G) = proportion of fixed point free elements.

How many?Let n = |Ω| > 1,G a transitive subgroup of Sym(Ω).δ(G) = proportion of fixed point free elements.Cameron-Cohen (1992): δ(G) ≥ 1/n with equality if and only if Gis sharply 2-transitive.Guralnick-Wan (1997): If n > 6 then δ(G) > 2/n unless G aFrobenius group of order n(n − 1)/2 or n(n − 1).

How many? IIFulman-Guralnick (2003): There exists δ > 0 such that δ(G) > δfor all finite nonabelian simple groups G.

How many? IIFulman-Guralnick (2003): There exists δ > 0 such that δ(G) > δfor all finite nonabelian simple groups G.Fulman-Guralnick (2003): There exists δ > 0 such thatδ(G) > δ/ log(n) for all primitive nonaffine groups G.

What are they?Fein-Kantor-Schacher (1981): G has a fixed point free element ofprime power order.

What are they?Fein-Kantor-Schacher (1981): G has a fixed point free element ofprime power order.Generalised Isbell ConjectureThere is a function f (p, k) such that if n = p a k with p̸ k anda ≥ f (p, k) then G has a fixed point free element of p-power order.

Elusive groupsWe say that a transitive permutation group is elusive if it has nofixed point free elements of prime order.

Elusive groupsWe say that a transitive permutation group is elusive if it has nofixed point free elements of prime order.For example, M 11 acting on 12 points.Existence of a fixed point free element of prime order is equivalentto existence of a semiregular subgroup.

Elusive groupsWe say that a transitive permutation group is elusive if it has nofixed point free elements of prime order.For example, M 11 acting on 12 points.Existence of a fixed point free element of prime order is equivalentto existence of a semiregular subgroup.G is elusive on Ω if and only if every conjugacy class of elements ofprime order meets G ω nontrivially.

A question of MarušičMarušič (1981): Are there any vertex-transitive digraphs with nofixed point free automorphisms of prime order?

A question of MarušičMarušič (1981): Are there any vertex-transitive digraphs with nofixed point free automorphisms of prime order?Independently posed by Jordan in 1988.

A question of MarušičMarušič (1981): Are there any vertex-transitive digraphs with nofixed point free automorphisms of prime order?Independently posed by Jordan in 1988.A digraph is a Cayley digraph if and only if its automorphism groupcontains a regular subgroup.

A question of MarušičMarušič (1981): Are there any vertex-transitive digraphs with nofixed point free automorphisms of prime order?Independently posed by Jordan in 1988.A digraph is a Cayley digraph if and only if its automorphism groupcontains a regular subgroup.Many methods for constructing Hamiltonian cycles use existence ofsuch an automorphism.

Nice representationsBiggs 197300 1100 1100 1100 1100 1100 115 | 100 11Petersen graph00 11 00 1100 11 00 1100 115 | 2

Nice representationsBiggs 197300 1100 1100 1100 1100 1100 115 | 100 11Petersen graph00 11 00 1100 11 00 1100 115 | 2000 11100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 11 00 1100 1100 1100 1100 1100 1100 11 00 1100 1100 1100 1100 1100 1100 1100 1100 117 | 100 11 00 1100 1100 117 | 47 | 2Coxeter graph

Nice representations II00110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011 0011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011 001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011 0011001100110011001100110011001100110011001100110011001100110011 00110011001100110011001100110011001100110011 00110011001117 | 2 17 | 117 | 417 | 8Biggs-Smith graph

2-closuresThe 2-closure G (2) of G is the group of all permutations of Ωwhich fix setwise each orbit of G on Ω × Ω.

2-closuresThe 2-closure G (2) of G is the group of all permutations of Ωwhich fix setwise each orbit of G on Ω × Ω.If G is 2-transitive on Ω then G (2) = Sym(Ω).

2-closuresThe 2-closure G (2) of G is the group of all permutations of Ωwhich fix setwise each orbit of G on Ω × Ω.If G is 2-transitive on Ω then G (2) = Sym(Ω).G (2) preserves all systems of imprimitivity for G.

2-closuresThe 2-closure G (2) of G is the group of all permutations of Ωwhich fix setwise each orbit of G on Ω × Ω.If G is 2-transitive on Ω then G (2) = Sym(Ω).G (2) preserves all systems of imprimitivity for G.We say that G is 2-closed if G = G (2) .

2-closuresThe 2-closure G (2) of G is the group of all permutations of Ωwhich fix setwise each orbit of G on Ω × Ω.If G is 2-transitive on Ω then G (2) = Sym(Ω).G (2) preserves all systems of imprimitivity for G.We say that G is 2-closed if G = G (2) .The full automorphism group of a digraph is 2-closed.

Polycirculant conjectureKlin (1997) extended the question of Marušič to 2-closed groups.

Polycirculant conjectureKlin (1997) extended the question of Marušič to 2-closed groups.Polycirculant ConjectureEvery finite transitive 2-closed permutation group has a fixed pointfree element of prime order.

Polycirculant conjectureKlin (1997) extended the question of Marušič to 2-closed groups.Polycirculant ConjectureEvery finite transitive 2-closed permutation group has a fixed pointfree element of prime order.In action on 12 points, (M 11 ) (2) = S 12 .

Early resultsMarušič (1981): All transitive permutation groups of degree p k ormp, for some prime p and m < p, have a fixed point free elementof order p.

Early resultsMarušič (1981): All transitive permutation groups of degree p k ormp, for some prime p and m < p, have a fixed point free elementof order p.Marušič and Scapellato (1993):• All cubic vertex-transitive graphs have a fixed point freeautomorphism of prime order.

Early resultsMarušič (1981): All transitive permutation groups of degree p k ormp, for some prime p and m < p, have a fixed point free elementof order p.Marušič and Scapellato (1993):• All cubic vertex-transitive graphs have a fixed point freeautomorphism of prime order.• A vertex-transitive digraph of order 2p 2 has a fixed point freeautomorphism of order p.

Fein-Kantor-Schacher examplesAGL(1, p 2 ), for p a Mersenne prime, acting on the set of p(p + 1)lines of the affine plane AG(2, p).

Fein-Kantor-Schacher examplesAGL(1, p 2 ), for p a Mersenne prime, acting on the set of p(p + 1)lines of the affine plane AG(2, p).All elements of order 2 and p fix a line so action is elusive.

Fein-Kantor-Schacher examplesAGL(1, p 2 ), for p a Mersenne prime, acting on the set of p(p + 1)lines of the affine plane AG(2, p).All elements of order 2 and p fix a line so action is elusive.2-closure contains C p+1pwhose p + 1 orbits are the parallel classes.AΓL(1, p 2 ) is also elusive in this action.

More constructionsCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)Suppose G 1 , G 2 are elusive groups on the sets Ω 1 , Ω 2 respectively.Then• G 1 × G 2 is elusive on Ω 1 × Ω 2 .• G 1 wr G 2 is elusive on Ω 1 × Ω 2 .

More constructionsCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)Suppose G 1 , G 2 are elusive groups on the sets Ω 1 , Ω 2 respectively.Then• G 1 × G 2 is elusive on Ω 1 × Ω 2 .• G 1 wr G 2 is elusive on Ω 1 × Ω 2 .• G 1 wr S n is elusive on Ω n .

General affine constructionCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)• V a vector space over a field of characteristic p,• G 1 GL(V ) with order prime to p,• W a subspace of V ,• H 1 < G 1 fixes W setwise.

General affine constructionCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)• V a vector space over a field of characteristic p,• G 1 GL(V ) with order prime to p,• W a subspace of V ,• H 1 < G 1 fixes W setwise.The action of V ⋊ G 1 on the set of right cosets of W ⋊ H 1 iselusive if and only if1 the images of W under G 1 cover V , and2 every conjugacy class of elements of prime order in G 1 meetsH 1 .

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1G 1 = C p 2 −1 GL(2, p), W a 1-dimensional subspace, H 1 = C p−1

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1G 1 = C p 2 −1 GL(2, p), W a 1-dimensional subspace, H 1 = C p−11 G 1 acts transitively on nontrivial elements of V .

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1G 1 = C p 2 −1 GL(2, p), W a 1-dimensional subspace, H 1 = C p−11 G 1 acts transitively on nontrivial elements of V .2 Only primes dividing p 2 − 1 are those dividing p − 1.

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1G 1 = C p 2 −1 GL(2, p), W a 1-dimensional subspace, H 1 = C p−11 G 1 acts transitively on nontrivial elements of V .2 Only primes dividing p 2 − 1 are those dividing p − 1.

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1G 1 = C p 2 −1 GL(2, p), W a 1-dimensional subspace, H 1 = C p−11 G 1 acts transitively on nontrivial elements of V .2 Only primes dividing p 2 − 1 are those dividing p − 1.In fact, any group C 2 p ⋊ G 1 , where G 1 C p 2 −1 contains theSylow 2-subgroup, is elusive.

Some more affine examples• V = GF(7 3 ), W a 2-dimensional subspace, G 1 = 3 1+2 ⋊ Q 8 ,H 1 = 3 2 ⋊ C 2 . (CGJKLMN 2002)

Some more affine examples• V = GF(7 3 ), W a 2-dimensional subspace, G 1 = 3 1+2 ⋊ Q 8 ,H 1 = 3 2 ⋊ C 2 . (CGJKLMN 2002)• p a Mersenne prime, V = GF(2) p2 −1 , W a hyperplane,G 1 = AΓL(1, p 2 ), H 1 = (C p ⋊ C p−1 ) × C 2 . (MG 2007)

Some more affine examples• V = GF(7 3 ), W a 2-dimensional subspace, G 1 = 3 1+2 ⋊ Q 8 ,H 1 = 3 2 ⋊ C 2 . (CGJKLMN 2002)• p a Mersenne prime, V = GF(2) p2 −1 , W a hyperplane,G 1 = AΓL(1, p 2 ), H 1 = (C p ⋊ C p−1 ) × C 2 . (MG 2007)• V = GF(2) 8 , W a codimension 2 space,G 1 = AΓL(1, 9), H 1 = (C 3 ⋊ C 2 ) × C 2 . (MG 2007)

Hensel lifting constructionCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)• V , G 1 , W and H 1 as in affine construction.• V dimension d over GF(p).• W dimension k.

Hensel lifting constructionCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)• V , G 1 , W and H 1 as in affine construction.• V dimension d over GF(p).• W dimension k.Then for any m > 1, the action of (C p m) d ⋊ G 1 on the set ofcosets of (C p m) k ⋊ H 1 is elusive of degree p km |G 1 : H 1 |.

Hensel lifting constructionCameron-MG-Jones-Kantor-Klin-Marušič-Nowitz (2002)• V , G 1 , W and H 1 as in affine construction.• V dimension d over GF(p).• W dimension k.Then for any m > 1, the action of (C p m) d ⋊ G 1 on the set ofcosets of (C p m) k ⋊ H 1 is elusive of degree p km |G 1 : H 1 |.This takes elusive groups of degree d and builds new ones withdegree p m d.

Priming ConstructionMG 2007Input:• V ⋊ 〈a〉 elusive of degree n,• 〈a〉 irreducible on V with order prime to p,• point stabiliser W ⋊ 〈b〉,• r a prime dividing |a|.

Priming ConstructionMG 2007Input:• V ⋊ 〈a〉 elusive of degree n,• 〈a〉 irreducible on V with order prime to p,• point stabiliser W ⋊ 〈b〉,• r a prime dividing |a|.Let• V ′ = V}⊕ ·{{· · ⊕ V}, W ′ = W ⊕ V ⊕ · · · ⊕ Vr copies• g = (1, . . . , 1, a)τ where τ = (12 . . . r)

Priming ConstructionMG 2007Input:• V ⋊ 〈a〉 elusive of degree n,• 〈a〉 irreducible on V with order prime to p,• point stabiliser W ⋊ 〈b〉,• r a prime dividing |a|.Let• V ′ = V}⊕ ·{{· · ⊕ V}, W ′ = W ⊕ V ⊕ · · · ⊕ Vr copies• g = (1, . . . , 1, a)τ where τ = (12 . . . r)Then V ′ ⋊ 〈g〉 acts elusively on the set of right cosets ofW ′ ⋊ 〈(b, . . . , b)〉 with degree rn.

Priming Construction IIUsing the FKS examples as a starting point, we get elusive groupsfor all degreesp2 n r j 11 . . . r tjtwhere• p Mersenne,• 2 n > p,• r 1 , . . . , r t are distinct odd primes dividing p − 1,• j i ≥ 0.

Priming Construction IIUsing the FKS examples as a starting point, we get elusive groupsfor all degreesp2 n r j 11 . . . r tjtwhere• p Mersenne,• 2 n > p,• r 1 , . . . , r t are distinct odd primes dividing p − 1,• j i ≥ 0.Using Hensel lifting construction can also multiply by powers of p.

A characterisationTheorem (MG-Kelly 2008+)Let G be an elusive permutation group such that G = N ⋊ G 1 forN an elementary abelian minimal normal subgroup and G 1 cyclic.Then G can be obtained by repeatedly applying the primingconstruction to an FKS-group.

Primitive groups and generalisationsLet G Sym(Ω) transitive• G is primitive if there are no nontrivial partitions of Ωpreserved by G.

Primitive groups and generalisationsLet G Sym(Ω) transitive• G is primitive if there are no nontrivial partitions of Ωpreserved by G.• G is quasiprimitive if all nontrivial normal subgroup of G aretransitive.

Primitive groups and generalisationsLet G Sym(Ω) transitive• G is primitive if there are no nontrivial partitions of Ωpreserved by G.• G is quasiprimitive if all nontrivial normal subgroup of G aretransitive.• G is biquasiprimitive if G is not quasiprimitive and allnontrivial normal subgroups have at most two orbits.

ClassificationsTheorem (MG 2003)The only almost simple elusive groups are M 11 and M 10 = A 6. 2acting on 12 points.

ClassificationsTheorem (MG 2003)The only almost simple elusive groups are M 11 and M 10 = A 6. 2acting on 12 points.Theorem (MG 2003)Let G be an elusive permutation group with a transitive minimalnormal subgroup. Then G = M 11 wr K acting on 12 k points withK a transitive subgroup of S k .

ClassificationsTheorem (MG 2003)The only almost simple elusive groups are M 11 and M 10 = A 6. 2acting on 12 points.Theorem (MG 2003)Let G be an elusive permutation group with a transitive minimalnormal subgroup. Then G = M 11 wr K acting on 12 k points withK a transitive subgroup of S k .2-closures of exceptions contain S k 12 .

ClassificationsTheorem (MG 2003)The only almost simple elusive groups are M 11 and M 10 = A 6. 2acting on 12 points.Theorem (MG 2003)Let G be an elusive permutation group with a transitive minimalnormal subgroup. Then G = M 11 wr K acting on 12 k points withK a transitive subgroup of S k .2-closures of exceptions contain S k 12 .All minimal normal subgroups of a counterexample to thepolycirculant conjecture must be intransitive.

Classifications IITheorem (MG-Xu 2007)Let G be a biquasiprimitive elusive permutation group on Ω. Thenone of the following holds:1 G = M 10 and |Ω| = 12;2 G = M 11 wr K and |Ω| = 2(12 k ), where K S k is transitivewith an index two subgroup;3 G = M 11 wr K and |Ω| = 2(12) k/2 , where K S k is transitivewith an index two intransitive subgroup.

Classifications IITheorem (MG-Xu 2007)Let G be a biquasiprimitive elusive permutation group on Ω. Thenone of the following holds:1 G = M 10 and |Ω| = 12;2 G = M 11 wr K and |Ω| = 2(12 k ), where K S k is transitivewith an index two subgroup;3 G = M 11 wr K and |Ω| = 2(12) k/2 , where K S k is transitivewith an index two intransitive subgroup.CorollaryLet G be a biquasiprimitive elusive permutation group on Ω. ThenG (2) is not elusive.

Locally quasiprimitive graphsΓ a graph, G Aut(Γ).

Locally quasiprimitive graphsΓ a graph, G Aut(Γ).Γ is G-locally quasiprimitive if for all vertices v, G v isquasiprimitive on Γ(v).

Locally quasiprimitive graphsΓ a graph, G Aut(Γ).Γ is G-locally quasiprimitive if for all vertices v, G v isquasiprimitive on Γ(v).A 2-arc in a graph is a triple (v 0 , v 1 , v 2 ) such that v 0 ∼ v 1 ∼ v 2and v 0 ≠ v 2 .

Locally quasiprimitive graphsΓ a graph, G Aut(Γ).Γ is G-locally quasiprimitive if for all vertices v, G v isquasiprimitive on Γ(v).A 2-arc in a graph is a triple (v 0 , v 1 , v 2 ) such that v 0 ∼ v 1 ∼ v 2and v 0 ≠ v 2 .Γ is 2-arc transitive if Aut(Γ) is transitive on the set of 2-arcs in Γ.

Locally quasiprimitive graphsΓ a graph, G Aut(Γ).Γ is G-locally quasiprimitive if for all vertices v, G v isquasiprimitive on Γ(v).A 2-arc in a graph is a triple (v 0 , v 1 , v 2 ) such that v 0 ∼ v 1 ∼ v 2and v 0 ≠ v 2 .Γ is 2-arc transitive if Aut(Γ) is transitive on the set of 2-arcs in Γ.A vertex-transitive graph is 2-arc-transitive if and only if G v is2-transitive on Γ(v) for all v.

Locally quasiprimitive graphs IITheorem (Praeger 1985)Let Γ be a finite connected G-vertex-transitive,G-locally-quasiprimitive graph and let 1 ≠ N ⊳ G. If N has atleast three vertex-orbits then it is semiregular.

Locally quasiprimitive graphs IITheorem (Praeger 1985)Let Γ be a finite connected G-vertex-transitive,G-locally-quasiprimitive graph and let 1 ≠ N ⊳ G. If N has atleast three vertex-orbits then it is semiregular.CorollaryEither G contains a semiregular subgroup, or G is quasiprimitive orbiquasiprimitive on vertices.

Locally quasiprimitive graphs IIITheorem (MG-Xu 2007)Every vertex-transitive, locally quasiprimitive graph has a fixedpoint free automorphism of prime order.

Locally quasiprimitive graphs IIITheorem (MG-Xu 2007)Every vertex-transitive, locally quasiprimitive graph has a fixedpoint free automorphism of prime order.CorollaryEvery 2-arc-transitive graph has a fixed point free automorphism ofprime order.

Locally quasiprimitive graphs IIITheorem (MG-Xu 2007)Every vertex-transitive, locally quasiprimitive graph has a fixedpoint free automorphism of prime order.CorollaryEvery 2-arc-transitive graph has a fixed point free automorphism ofprime order.CorollaryEvery arc-transitive graph of prime valency has a fixed point freeautomorphism of prime order.

Some more resultsDobson-Malnič-Marušič-Nowitz (2007):• All quartic vertex-transitive graphs have a fixed point freeautomorphism of prime order.

Some more resultsDobson-Malnič-Marušič-Nowitz (2007):• All quartic vertex-transitive graphs have a fixed point freeautomorphism of prime order.• All vertex-transitive graphs of valency p + 1 admitting atransitive {2, p}-group for p odd have a fixed point freeautomorphism of prime order.

Some more resultsDobson-Malnič-Marušič-Nowitz (2007):• All quartic vertex-transitive graphs have a fixed point freeautomorphism of prime order.• All vertex-transitive graphs of valency p + 1 admitting atransitive {2, p}-group for p odd have a fixed point freeautomorphism of prime order.• There are no elusive 2-closed groups of square-free degree.

Some more resultsDobson-Malnič-Marušič-Nowitz (2007):• All quartic vertex-transitive graphs have a fixed point freeautomorphism of prime order.• All vertex-transitive graphs of valency p + 1 admitting atransitive {2, p}-group for p odd have a fixed point freeautomorphism of prime order.• There are no elusive 2-closed groups of square-free degree.Xu (2008): All arc-transitive graphs with valency pq, p, q primes,such that Aut(Γ) has a nonabelian minimal normal subgroup Nwith at least 3 vertex orbits, has a semiregular automorphism.

Open problems and questions• Prove the polycirculant conjecture.

Open problems and questions• Prove the polycirculant conjecture.• Prove the polycirculant conjecture for arc-transitive graphs.

Open problems and questions• Prove the polycirculant conjecture.• Prove the polycirculant conjecture for arc-transitive graphs.• Find new constructions of elusive groups.

Open problems and questions• Prove the polycirculant conjecture.• Prove the polycirculant conjecture for arc-transitive graphs.• Find new constructions of elusive groups.• For what degrees do elusive groups exist? (smallest degree forwhich existence is unknown is 40.)

Open problems and questions• Prove the polycirculant conjecture.• Prove the polycirculant conjecture for arc-transitive graphs.• Find new constructions of elusive groups.• For what degrees do elusive groups exist? (smallest degree forwhich existence is unknown is 40.)• Does the set of all such degrees have density 0?

Open problems and questions• Prove the polycirculant conjecture.• Prove the polycirculant conjecture for arc-transitive graphs.• Find new constructions of elusive groups.• For what degrees do elusive groups exist? (smallest degree forwhich existence is unknown is 40.)• Does the set of all such degrees have density 0?http://www.maths.uwa.edu.au/∼giudici/research.html

Group Theory, Combinatorics and ComputationAMSI theme program at The University of Western Australia5 th − 16 th January 2009First week international conference in honour of Cheryl Praeger’s60 th birthday.Informal second week consisting of short courses and problemsessions.For more information, seehttp://sponsored.uwa.edu.au/gcc09/welcome