Subdegrees of primitive permutation groups Michael Giudici Centre - PowerPoint PPT Presentation

Subdegrees of primitive permutation groups Michael Giudici Centre for the Mathematics of Symmetry and Computation G2D2 2019, Yichang, China

Subdegrees G � Sym(Ω) transitive on a finite set Ω. Let α ∈ Ω and G α be the stabiliser of α ∈ G . An orbit Σ of G α on Ω is called a suborbit of G and | Σ | is called a subdegree. We call the suborbits other than { α } and their corresponding subdegrees nontrivial. Since G is transitive, the list of subdegrees is independent of the choice of α .

Examples • G = S n acting on n points: G α = S n − 1 . Subdegrees are 1 , n − 1 . • G = S n wr S 2 acting on 2 n points: G α = S n − 1 × S n . Subdegrees are 1 , n , n − 1 . • G = D 2 n for n even acting on n points: G α = C 2 . Subdegrees are 1 , 1 , 2 , . . . , 2 � �� � n − 2 times 2 • G = S 7 acting on the 21 two-subsets of a set of size 7. Subdegrees are 1 , 10 , 10 .

Orbital (di)graphs An orbital of G is an orbit of G on Ω × Ω. The number of orbitals is called the rank of G . There is a one-to-one correspondence between the orbitals of G and the suborbits of G α via ( α, β ) G ← → β G α Each orbital of G gives rise to an orbital digraph whose vertex set is Ω and arc set is ( α, β ) G . Every arc-transitive digraph arises in this way. If ( β, α ) ∈ ( α, β ) G then we call the orbital (and corresponding suborbit) self-paired and we can consider the orbital digraph as an undirected graph.

Example For all d � 2 there exists a transitive permutation group with d as a non-self-paired subdegree. Take G = S d wr C m acting on md points and G α = S d − 1 × S m − 1 . d ( d , m ) = (3 , 2) :

Primitive groups A transitive permutation group G on Ω is called imprimitive if G preserves some nontrivial partition of Ω. If no such partition exists then G is called primitive Lemma If G is primitive and β is fixed by G α then either β = α or G = C p acting on p points.

What subdegrees occur for primitive groups? • Constant subdegrees • Coprime subdegrees • Small subdegrees

Constant subdegrees G is 2-transitive if and only if G α is transitive on Ω \{ α } , that is, only one nontrivial subdegree. G is 3 2 -transitive if and only if all nontrivial suborbits have the same length. Wielandt (1964): A 3 2 -transitive is either a Frobenius group or primitive. Passman (1967): Classified the soluble examples.

Classification of 3 2 -transitive groups Bamberg, Giudici, Liebeck, Praeger, Saxl, Tiep (2013,2016,2019+) Let G be a 3 2 -transitive group on n points. Then one of the following holds: • G is 2-transitive; • G is a Frobenius group; • n = 21 and G = A 7 or S 7 ; • n = 2 f − 1 (2 f − 1) and either G = PSL 2 (2 f ), or P Γ L 2 (2 f ) with f a prime. • G = C d p ⋊ G 0 � AGL ( d , p ) and one of : • G 0 � Γ L 1 ( p d ); • G 0 = S 0 ( p d / 2 ) with p odd; • G 0 is soluble and p d = 3 2 , 5 2 , 7 2 , 11 2 , 17 2 or 3 4 ; • SL 2 (5) ⊳ G 0 � Γ L 2 ( p d / 2 ) with p d / 2 = 9 , 11 , 19 , 29 or 169.

Key steps • G is almost simple or affine. • If p divides | Ω | and there is a subdegree divisible by p then G is not 3 2 -transitive. • If G is a group of Lie type of characteristic p and p does not divide | G : G α | then G α is a parabolic. • Classify primitive actions of groups of Lie type of characteristic p with no subdegree divisible by p . • Classify primitive groups C d p ⋊ G 0 � AGL ( d , p ) with p dividing | G 0 | and no subdegrees divisible by p . • Classify insoluble possibilities for G 0 with | G 0 | coprime to p .

Coprime subdegrees Marie Weiss (1935): Let G be primitive. • If G has coprime nontrivial subdegrees d 1 > d 2 then G has a subdegree dividing d 1 d 2 and greater than d 1 . • If G has k pairwise coprime nontrivial subdegrees then G has rank at least 2 k . Note: • J 1 acting on 266 points has subdegrees 1, 11, 12, 110, 132. • d 1 d 2 need not be a subdegree. For example, G = HS acting on 3850 points with G α = 2 4 . S 6 has subdegrees: 1 , 15 , 32 , 90 , 120 , 160 , 192 , 240 , 240 , 360 , 960 , 1440

Number of pairwise coprime subdegrees Dolfi-Guralnick-Praeger-Spiga (2013, 2016): • The largest size of a set of pairwise coprime nontrivial subdegrees of a finite primitive group is at most 2. • If a primitive permutation group G has a pair of coprime subdegrees then G is almost simple, product action or twisted wreath type.

Examples • Let G = PSL 2 ( p ) for p ≡ ± 1 (mod 16) and p ≡ ± 1 (mod 5). Let G α = A 5 . Then G has 5 and 12 as subdegrees. • Suppose that H acts on ∆ and H δ has a suborbit γ H δ of size d . Let G = H wr S k act on ∆ k . Then letting α = ( δ, . . . , δ ) and β = ( γ, . . . , γ ) we have that G α = H δ wr S k and | β G α | = d k . Only gave one twisted wreath example.

Twisted Wreath Products Let T = PSL 2 ( q ) and G ( m , q ) = N ⋊ P for m � 2 where • N = T k with k = | T | m − 1 ; • P = T wr S m acts transitively on the k simple direct factors of N with P 1 = T × S m . Then G ( m , q ) acts primitively on the set of right cosets of P with N as a regular minimal normal subgroup. Chua-Giudici-Morgan (2019+): G ( m , q ) has a pair of coprime nontrivial subdegrees if and only if one of the following hold: • q ≡ 3 (mod 4) or q = 29, or • q is even and m � 3. 2(12) 2 and 55 2 G (2 , 11) has two coprime pairs: 11 2 and a divisor of 2(60 2 ).

Small subdegrees Lemma If G is primitive with 2 as a subdegree then G ∼ = D 2 p acting on p points with p a prime. Proof: Suppose that | β G α | = 2. Then | G α : G α,β | = 2 = | G β : G α,β | and G = � G α , G β � . Thus G α,β ⊳ G and so G α,β = 1. Thus | G α | = 2 and so G is generated by involutions. Thus G ∼ = D 2 n acting on n points and primitively implies n = p . Not true in infinite case.

Subdegree 3 Suppose that G is primitive on n points with 3 as a subdegree. Sims (1967): | G α | divides 3 . 2 4 . Wong (1967): G is one of the following: • C p ⋊ C 3 for p ≡ 1 (mod 3) with n = p ; • C 2 p ⋊ C 3 for p ≡ 2 (mod 3) with n = p 2 ; • C 2 p ⋊ S 3 for p ≥ 5 and n = p 2 ; • A 5 or S 5 with n = 10; • PGL 2 (7) with n = 28; • PSL 2 (11) and PSL 2 (13) acting on cosets of a D 12 ; • PSL 2 ( p ) with p ≡ ± 1 (mod 16) acting on cosets of an S 4 ; • SL 3 (3) or Aut( SL 3 (3)) with n = 234.

Subdegree 4 All primitive groups with 4 as a subdegree were classified by Wang (1992) following earlier work of Sims (1967) and Quirin (1971). Li-Lu-Maruˇ siˇ c (2004): Classified all vertex-primitive graphs of valency 3 or 4.

Subdegree 5 Wang (1995,1996): Investigated primitive permutation groups with 5 as a subdegree. Classified the cases where G v is soluble or G v is unfaithful. Fawcett-Giudici-Li-Praeger-Royle-Verret (2018): • Complete classification: 14 infinite families and 13 sporadic examples. • Classified all almost simple groups with A 5 or S 5 as a maximal subgroup (using results of David Craven for exceptional groups of Lie type). • Classified all vertex-primitive graphs of valency 5

Primitive groups with 5 as a subdegree: sporadic examples G G v | G : G v | Alt(5) D 10 6 Sym(5) AGL (1 , 5) 6 PGL (2 , 9) 36 D 20 M 10 AGL (1 , 5) 36 P Γ L (2 , 9) AGL (1 , 5) × Z 2 36 PGL (2 , 11) D 20 66 Alt(9) (Alt(4) × Alt(5)) ⋊ Z 2 126 Sym(9) Sym(4) × Sym(5) 126 PSL (2 , 19) 171 D 20 Suz(8) AGL (1 , 5) 1 456 J 3 AGL (2 , 4) 17 442 J 3 ⋊ Z 2 A Γ L (2 , 4) 17 442 Th Sym(5) 756 216 199 065 600

Primitive groups with 5 as a subdegree: infinite families G G v | G : G v | Conditions Z p ⋊ Z 5 Z 5 p p ≡ 1 (mod 5) Z 2 p 2 p ⋊ Z 5 Z 5 p ≡ − 1 (mod 5) Z 4 p 4 p ⋊ Z 5 Z 5 p ≡ ± 2 (mod 5) Z 2 p 2 p ⋊ D 10 D 10 p ≡ ± 1 (mod 5) Z 4 p 4 p ≡ ± 2 (mod 5) p ⋊ D 10 D 10 Z 4 p 4 p ⋊ AGL (1 , 5) AGL (1 , 5) p � = 5 Z 4 p 4 p ⋊ Alt(5) Alt(5) p � = 5 Z 4 p 4 p ⋊ Sym(5) Sym(5) p � = 5 p 3 − p PSL (2 , p ) Alt(5) p ≡ ± 1 , ± 9 (mod 40) 120 p 6 − p 2 PSL (2 , p 2 ) Alt(5) p ≡ ± 3 (mod 10) 120 p 6 − p 2 P Σ L (2 , p 2 ) Sym(5) p ≡ ± 3 (mod 10) 120 p 9 ( p 6 − 1)( p 4 − 1)( p 2 − 1) PSp (6 , p ) Sym(5) p ≡ ± 1 (mod 8) 240 p 9 ( p 6 − 1)( p 4 − 1)( p 2 − 1) PSp (6 , p ) Alt(5) p ≡ ± 3 , ± 13 (mod 40) 120 p 9 ( p 6 − 1)( p 4 − 1)( p 2 − 1) PGSp (6 , p ) Sym(5) p ≡ ± 3 (mod 8), p � 11 120

Vertex-primitive graphs of valency 5 Aut(Γ) Aut(Γ) v | V (Γ) | Conditions Z 4 2 ⋊ Sym(5) Sym(5) 16 P Γ L (2 , 9) AGL (1 , 5) × Z 2 36 PGL (2 , 11) 66 D 20 Sym(9) Sym(4) × Sym(5) 126 Suz(8) AGL (1 , 5) 1 456 J 3 ⋊ 2 A Γ L (2 , 4) 17 442 Th Sym(5) 756 216 199 065 600 p 3 − p PSL (2 , p ) Alt(5) p ≡ ± 1 , ± 9 (mod 40) 120 p 6 − p 2 P Σ L (2 , p 2 ) Sym(5) p ≡ ± 3 (mod 10) 120 p 9 ( p 6 − 1)( p 4 − 1)( p 2 − 1) PSp (6 , p ) Sym(5) p ≡ ± 1 (mod 8) 240 p 9 ( p 6 − 1)( p 4 − 1)( p 2 − 1) PGSp (6 , p ) Sym(5) p ≡ ± 3 (mod 8) 120 p � 11

Vertex-primitive half-arc-transitive graphs A graph Γ is called half-arc-transitive if its automorphism group G = Aut(Γ) is transitive on edges but not on arcs. Such a graph has valency 2 d and G has two paired suborbits of length d . Li-Lu-Maruˇ siˇ c (2004): • There are no vertex-primitive half arc-transitive graphs of valency less than 10. • There exists a vertex-primitive half-arc-transitive graph of valency 14. FGLPRV (2018): The smallest valency of a vertex-primitive half-arc-transitive graph is 12.