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 Gα be the stabiliser of α ∈ G. An orbit Σ of Gα 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 = Sn acting on n points: Gα = Sn−1. Subdegrees are

1, n − 1.

• G = Sn wr S2 acting on 2n points: Gα = Sn−1 × Sn.

Subdegrees are 1, n, n − 1.

• G = D2n for n even acting on n points: Gα = C2.

Subdegrees are 1, 1, 2, . . . , 2

n−2 2

times

• G = S7 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 Gα via (α, β)G ← → βGα 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 = Sd wr Cm acting on md points and Gα = Sd−1 × Sm−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 Gα then either β = α

• r G = Cp 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 Gα 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 = A7 or S7;
• n = 2f −1(2f − 1) and either G = PSL2(2f ), or PΓL2(2f )

with f a prime.

• G = C d

p ⋊ G0 AGL(d, p) and one of :

• G0 ΓL1(pd);
• G0 = S0(pd/2) with p odd;
• G0 is soluble and pd = 32, 52, 72, 112, 172 or 34;
• SL2(5) ⊳ G0 ΓL2(pd/2) with pd/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 : Gα| then Gα 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 ⋊ G0 AGL(d, p) with p

dividing |G0| and no subdegrees divisible by p.

• Classify insoluble possibilities for G0 with |G0| coprime

to p.

Coprime subdegrees

Marie Weiss (1935): Let G be primitive.

• If G has coprime nontrivial subdegrees d1 > d2 then G has

a subdegree dividing d1d2 and greater than d1.

• If G has k pairwise coprime nontrivial subdegrees then G

has rank at least 2k. Note:

• J1 acting on 266 points has subdegrees 1, 11, 12, 110, 132.
• d1d2 need not be a subdegree. For example, G = HS

acting on 3850 points with Gα = 24.S6 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 = PSL2(p) for p ≡ ±1 (mod 16) and p ≡ ±1

(mod 5). Let Gα = A5. Then G has 5 and 12 as subdegrees.

• Suppose that H acts on ∆ and Hδ has a suborbit γHδ of

size d. Let G = H wr Sk act on ∆k. Then letting α = (δ, . . . , δ) and β = (γ, . . . , γ) we have that Gα = Hδ wr Sk and |βGα| = dk. Only gave one twisted wreath example.

Twisted Wreath Products

Let T = PSL2(q) and G(m, q) = N ⋊ P for m 2 where

• N = T k with k = |T|m−1;
• P = T wr Sm acts transitively on the k simple direct

factors of N with P1 = T × Sm. 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.

G(2, 11) has two coprime pairs: 2(12)2 and 552 112 and a divisor of 2(602).

Small subdegrees

Lemma If G is primitive with 2 as a subdegree then G ∼ = D2p acting on p points with p a prime. Proof: Suppose that |βGα| = 2. Then |Gα : Gα,β| = 2 = |Gβ : Gα,β| and G = Gα, Gβ. Thus Gα,β ⊳ G and so Gα,β = 1. Thus |Gα| = 2 and so G is generated by involutions. Thus G ∼ = D2n 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): |Gα| divides 3.24. Wong (1967): G is one of the following:

• Cp ⋊ C3 for p ≡ 1 (mod 3) with n = p;
• C 2

p ⋊ C3 for p ≡ 2 (mod 3) with n = p2;

• C 2

p ⋊ S3 for p ≥ 5 and n = p2;

• A5 or S5 with n = 10;
• PGL2(7) with n = 28;
• PSL2(11) and PSL2(13) acting on cosets of a D12;
• PSL2(p) with p ≡ ±1 (mod 16) acting on cosets of an S4;
• SL3(3) or Aut(SL3(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 Gv is soluble

• r Gv is unfaithful.

Fawcett-Giudici-Li-Praeger-Royle-Verret (2018):

• Complete classification: 14 infinite families and 13

sporadic examples.

• Classified all almost simple groups with A5 or S5 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 Gv |G : Gv| Alt(5) D10 6 Sym(5) AGL(1, 5) 6 PGL(2, 9) D20 36 M10 AGL(1, 5) 36 PΓL(2, 9) AGL(1, 5) × Z2 36 PGL(2, 11) D20 66 Alt(9) (Alt(4) × Alt(5)) ⋊ Z2 126 Sym(9) Sym(4) × Sym(5) 126 PSL(2, 19) D20 171 Suz(8) AGL(1, 5) 1 456 J3 AGL(2, 4) 17 442 J3 ⋊ Z2 AΓL(2, 4) 17 442 Th Sym(5) 756 216 199 065 600

Primitive groups with 5 as a subdegree: infinite families

G Gv |G : Gv| Conditions Zp ⋊ Z5 Z5 p p ≡ 1 (mod 5) Z2

p ⋊ Z5

Z5 p2 p ≡ −1 (mod 5) Z4

p ⋊ Z5

Z5 p4 p ≡ ±2 (mod 5) Z2

p ⋊ D10

D10 p2 p ≡ ±1 (mod 5) Z4

p ⋊ D10

D10 p4 p ≡ ±2 (mod 5) Z4

p ⋊ AGL(1, 5)

AGL(1, 5) p4 p = 5 Z4

p ⋊ Alt(5)

Alt(5) p4 p = 5 Z4

p ⋊ Sym(5)

Sym(5) p4 p = 5 PSL(2, p) Alt(5)

p3−p 120

p ≡ ±1, ±9 (mod 40) PSL(2, p2) Alt(5)

p6−p2 120

p ≡ ±3 (mod 10) PΣL(2, p2) Sym(5)

p6−p2 120

p ≡ ±3 (mod 10) PSp(6, p) Sym(5)

p9(p6−1)(p4−1)(p2−1) 240

p ≡ ±1 (mod 8) PSp(6, p) Alt(5)

p9(p6−1)(p4−1)(p2−1) 120

p ≡ ±3, ±13 (mod 40) PGSp(6, p) Sym(5)

p9(p6−1)(p4−1)(p2−1) 120

p ≡ ±3 (mod 8), p 11

Vertex-primitive graphs of valency 5

Aut(Γ) Aut(Γ)v |V (Γ)| Conditions Z4

2 ⋊ Sym(5)

Sym(5) 16 PΓL(2, 9) AGL(1, 5) × Z2 36 PGL(2, 11) D20 66 Sym(9) Sym(4) × Sym(5) 126 Suz(8) AGL(1, 5) 1 456 J3 ⋊ 2 AΓL(2, 4) 17 442 Th Sym(5) 756 216 199 065 600 PSL(2, p) Alt(5)

p3−p 120

p ≡ ±1, ±9 (mod 40) PΣL(2, p2) Sym(5)

p6−p2 120

p ≡ ±3 (mod 10) PSp(6, p) Sym(5)

p9(p6−1)(p4−1)(p2−1) 240

p ≡ ±1 (mod 8) PGSp(6, p) Sym(5)

p9(p6−1)(p4−1)(p2−1) 120

p ≡ ±3 (mod 8) 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 2d 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.

Construction

Let G = PSp6(p) such that p ≡ 7, 23 (mod 40). Let M ∼ = S5 be a maximal subgroup with an index 6 subgroup H. Then NG(H)/H ∼ = Cp+1 so get a non-self-paired suborbit of length 6. Let Γ be the underlying graph of the orbital digraph. Then Aut(Γ) = G.