A generalization of Sims’ conjecture for finite primitive groups and two point stabilizers in primitive groups

Theorem 1.1 (Sims conjecture)

There is a function f : N → N such that, if 𝐺 is a finite primitive group with a suborbit of length d > 1 , then the stabilizers have order at most f ⁢ ( d ) .

There exists a function g : N → N such that, if 𝐺 is a finite primitive group and 𝛼 and 𝛽 are two distinct points in the domain of 𝐺, then either

1. G α ⁢ β has order at most g ( | G α ⁢ β : G α + [ 1 ] | ) , or

2. 𝐺 is in a well-described and well-determined list of exceptions.

Let 𝐺 be a primitive group on Ω of SD type, let 𝛼 and 𝛽 be two distinct elements from Ω, and consider the action of 𝐺 on the orbital graph determined by ( α , β ) . Then one of the following holds:

• G α + [ 1 ] = 1 ,

• | G α + [ 1 ] | = ℓ + 1 , G α + [ 1 ] ≤ Sym ⁡ ( ℓ + 1 ) and the group G α + [ 1 ] acts regularly on the set { 1 , … , ℓ + 1 } . Moreover, let

  ( t 0 , t 1 , … , t ℓ ) ∈ N   with ⁢ t 0 = 1   and   β = D ⁢ ( t 0 , t 1 , … , t ℓ ) .

  The mapping G α + [ 1 ] → T defined by σ → t 0 σ - 1 is a group homomorphism.

There exists a finite primitive group 𝐺 and two distinct elements 𝛼 and 𝛽 in the domain of 𝐺 such that G α ⁢ β ⊴ G α and G α ⁢ β ≠ 1 .

Let 𝐺 be a transitive group on Ω containing a normal regular subgroup, let Γ be a connected digraph with vertex set Ω that is left invariant by the action of 𝐺. Then G α + [ 1 ] = 1 , for every α ∈ Ω . In particular, Conjecture 1.2 (i) holds true when 𝐺 has O’Nan–Scott type HA , TW , HS and HC and the function 𝑔 can be taken so that g ⁢ ( n ) := n for every n ∈ N .

Proof

Let α ∈ Ω . Let 𝑁 be a regular normal subgroup of 𝐺, and let H := G α . Then 𝐺 is the semidirect product of 𝑁 by 𝐻 (that is, G = N ⁢ G α , N ∩ G α = 1 and G = N ⋊ H ), and the action of 𝐺 on Ω is permutation equivalent to the “affine” action of 𝐺 on 𝑁, where 𝑁 acts on 𝑁 by right multiplication and where 𝐻 acts on 𝑁 by conjugation. In the rest of the proof, we use this identification. In particular, under this equivalence, α ∈ Ω corresponds to 1 ∈ N . For each β ∈ Γ + ⁢ ( α ) , we let n β be the element of 𝑁 corresponding to 𝛽, that is, α n β = β .

Then G α ⁢ β = C H ⁢ ( n β ) and

Since Γ is connected, we deduce N = ⟨ n β ∣ β ∈ Γ + ⁢ ( α ) ⟩ . Hence

In other words, G α acts faithfully on Γ + ⁢ ( α ) .

When 𝐺 has O’Nan–Scott type HA , TW , HS and HC , the proof follows by the fact that 𝐺 contains a normal regular subgroup and by the fact that the non-trivial orbital graphs of 𝐺 are connected. ∎

Proof of Theorem 1.3

We use the notation that we have established above. Without loss of generality, we may assume that α = D ⁢ ( 1 , 1 , … , 1 ) . Write

for some t 1 , … , t ℓ ∈ T . We set t 0 := 1 , in particular, β = D ⁢ ( t 0 , t 1 , … , t ℓ ) . This notation will make the last part of our proof easier to follow.

Let φ ∈ G α + [ 1 ] ∩ Aut ⁡ ( T ) . For each t ∈ T , we let ι t ∈ Aut ⁡ ( T ) ≤ W α denote the permutation on Ω induced by the conjugation via 𝑡. Observe that we have ι t ∈ G ∩ W α = G α for every t ∈ T because T ℓ + 1 = N ≤ G . As φ ∈ G α + [ 1 ] and ι t ∈ G α , we deduce that 𝜑 fixes β ι t for every t ∈ T . This means that

for every t ∈ T . Since β ≠ α , there exists i ∈ { 1 , … , ℓ } with t i ≠ 1 . Now, (4.1) gives t i t ⁢ φ = t i t for every t ∈ T . Therefore,

This shows that

Now, G α + [ 1 ] is a normal subgroup of G α . Since G α acts primitively as a group of permutations on the ℓ + 1 simple direct factors of T ℓ + 1 , we obtain that either G α + [ 1 ] projects trivially on Sym ⁡ ( ℓ + 1 ) or G α + [ 1 ] projects to a transitive subgroup of Sym ⁡ ( ℓ + 1 ) . If G α + [ 1 ] projects trivially on Sym ⁡ ( ℓ + 1 ) , then G α + [ 1 ] ≤ Aut ⁡ ( T ) and hence G α + [ 1 ] = 1 by (4.2). (In particular, in this case, Conjecture 1.2 part (i) holds true). Therefore, for the rest of this proof, we assume that

Observe now that

As G α + [ 1 ] and ι ⁢ ( T ) are both normal in G α , we deduce that G α + [ 1 ] centralizes ι ⁢ ( T ) . Since C W α ⁢ ( ι ⁢ ( T ) ) = Sym ⁡ ( ℓ + 1 ) , we get G α + [ 1 ] ≤ Sym ⁡ ( ℓ + 1 ) . Therefore, G α + [ 1 ] is a transitive subgroup of Sym ⁡ ( ℓ + 1 ) .

Next we show that the group G α + [ 1 ] is a regular subgroup of Sym ⁡ ( ℓ + 1 ) . Let H := N G ⁢ ( T 0 ) , where T 0 is the first simple direct factor of the socle 𝑁. Now, 𝐻 acts transitively on Ω because N ≤ H , and 𝐻 contains the normal regular subgroup M = T 1 × ⋯ × T ℓ . Therefore, by Proposition 3.1 applied to 𝐻, we deduce H α + [ 1 ] = 1 . As | G : H | = | G : N G ( T 0 ) | = ℓ + 1 and H α + [ 1 ] = H ∩ G α + [ 1 ] , we get | G α + [ 1 ] | ≤ ℓ + 1 . Since G α + [ 1 ] is a transitive subgroup of Sym ⁢ ( ℓ + 1 ) , G α + [ 1 ] is a regular subgroup of Sym ⁡ ( ℓ + 1 ) and ℓ + 1 = | G α + [ 1 ] | .

We need to recall in detail the action of Sym ⁡ ( ℓ + 1 ) on Ω. Given

we have

The element 𝜎 in the right-hand side of (4.3) appears as σ - 1 to guarantee that this is a right action.

Recall β = D ⁢ ( 1 , t 1 , … , t ℓ ) = D ⁢ ( t 0 , t 1 , … , t ℓ ) . We now define a mapping

In other words, in the light of (4.3), w ⁢ ( σ ) is the first coordinate of

Let σ , τ ∈ G α + [ 1 ] . Since 𝜏 fixes 𝛽, we have

and since σ ⁢ τ fixes 𝛽, we have also

In other words, the two ( ℓ + 1 ) -tuples

differ only by the left multiplication by an element of 𝐷. Therefore, there exists t ∈ T such that

By checking the first coordinates in (4.4), we obtain

Moreover, by comparing the coordinate appearing in position 0 τ on the left-hand side and on the right-hand side of (4.4), we deduce that t ⁢ t ( 0 τ ) τ - 1 = t ( 0 τ ) ( σ ⁢ τ ) - 1 , that is,

Putting these two equations together, we obtain

This proves that our mapping w : G α + [ 1 ] → T is a group homomorphism. ∎

Let 𝑝 be a prime number, let k ≥ 7 be a positive integer, and let 𝑟 be a primitive prime divisor of p k - 1 , that is, 𝑟 divides p k - 1 and 𝑟 is relatively prime to p i - 1 for each i ∈ { 1 , … , k - 1 } . As k ≥ 7 , the existence of 𝑟 is guaranteed by Zsigmondy’s theorem.

Let H := V ⋊ C be the affine primitive group of degree p k , where 𝑉 is an elementary abelian 𝑝-group of order p k and where 𝑅 is a cyclic group of order 𝑟. (We use an additive notation for 𝑉.) Let 𝑇 be a non-abelian simple group containing a cyclic subgroup 𝑃 of order 𝑝 and with C T ⁢ ( P ) = P , and let w : V → P be an arbitrary surjective homomorphism.

We denote by T V the set of all functions from 𝑉 to 𝑇. Observe that T V is a group isomorphic to the Cartesian product of | V | = p k copies of 𝑇. We denote the elements of T V as functions f : V → T .

We let 𝐺 be the primitive group of diagonal type T V ⋊ H . Recall that the elements of Ω are right cosets of 𝐷 in T V , where 𝐷 is the diagonal subgroup of T V , that is, D = { f ∈ T V ∣ f ⁢ is constant } . Let b = ( t v ) v ∈ V ∈ T V , where t v = ω ⁢ ( - v ) for every v ∈ V .

Let α := D , let β := D ⁢ b , and consider the orbital graph determined by ( α , β ) . Let v ∈ V . Then

Thus b v = w ⁢ ( v ) ⁢ b and hence β v = D ⁢ b v = D ⁢ b = β . This shows V ≤ G α ⁢ β . Since V ⊴ G α , we deduce that G α + [ 1 ] ≤ V . From Proposition 1.3, we have G α + [ 1 ] = V . Thus G α = ( T × R ) ⁢ G α + [ 1 ] and

Let φ := t ⁢ h ∈ G α ⁢ β ∩ ( T × R ) , with t ∈ T and h ∈ R . Observe that

Since D ⁢ b = β = β t ⁢ h = D ⁢ b t ⁢ h = D ⁢ b , from (4.5), we get b t ⁢ h = b .

For every v ∈ Ker ⁢ ( w ) ≤ V , we have

Therefore, w ⁢ ( v h - 1 ) = 1 . This gives Ker ⁢ ( w ) h - 1 = Ker ⁢ ( w ) . As

and as 𝑅 is a cyclic group of prime order acting irreducibly on the vector space 𝑉, we deduce that h = 1 . This shows that φ = t ∈ T .

Now, b t = b if and only if 𝑡 centralizes all the coordinates of 𝑏. In other words, t ∈ C T ⁢ ( P ) = P . Summing up,

Thus | G α ⁢ β : G α + [ 1 ] | = | P | = p and | G α + [ 1 ] | = p k . However, we cannot bound the cardinality of G α + [ 1 ] with 𝑝 only.