Multiple transitivity except for a system of imprimitivity

Let Ω be a set, let G ≤ Sym ⁢ ( Ω ) be a transitive group preserving the nonuniversal equivalence relation ∼ and let ω ∈ Ω . Then 𝐺 is 2-by-block-transitive if and only if the following equation is satisfied for some (or equivalently for all) g ∈ G ∖ G ⁢ ( [ ω ] ) :

Proof

Since 𝐺 acts transitively on Ω, it follows that there is a bijection 𝜃 from the set of ( G ⁢ ( ω ) , G ⁢ ( ω ) ) -double cosets in 𝐺 to the set of G ⁢ ( ω ) -orbits on Ω, where given x ∈ G , we set

Note that G ⁢ ( [ ω ] ) is itself a union of ( G ⁢ ( ω ) , G ⁢ ( ω ) ) -double cosets, and the G ⁢ ( [ ω ] ) -orbit of 𝜔 is exactly [ ω ] ; thus ⋃ { θ ⁢ ( G ⁢ ( ω ) ⁢ x ⁢ G ⁢ ( ω ) ) ∣ x ∈ G ⁢ ( [ ω ] ) } is the union of all G ⁢ ( ω ) -orbits on [ ω ] .

In particular, the double coset equation in the statement is equivalent to the assertion that G ⁢ ( ω ) acts transitively on Ω ∖ [ ω ] . In turn, we see that G ⁢ ( ω ) acts transitively on Ω ∖ [ ω ] if and only if 𝐺 acts transitively on distant pairs. ∎

Let Ω be a set and let 𝐺 be a finite group acting transitively on Ω and preserving the equivalence relation ∼ . Let ( ω , ω ′ ) ∈ Ω [ 2 ] . Then 𝐺 acts 2-by-block-transitively on Ω if and only if

In particular, if 𝐺 acts 2-by-block-transitively on Ω, then the right-hand side of the above equation is an integer.

Proof

Since 𝐺 is transitive, we can write ω ′ = g ⁢ ω for some g ∈ G ; note that g ∉ G ⁢ ( [ ω ] ) . We can calculate the size of the double coset G ⁢ ( ω ) ⁢ g ⁢ G ⁢ ( ω ) as

so the equation in the statement is equivalent to the equation

By Lemma 2.1, the latter equation is equivalent to 𝐺 acting 2-by-block-transitively on Ω. ∎

Let Ω be a set and let G ≤ Sym ⁢ ( Ω ) be 𝑘-by-block-transitive on Ω relative to the equivalence relation ∼ , for k ≥ 2 . Then, for each ω ∈ Ω , the block stabilizer G ⁢ ( [ ω ] ) is the largest proper subgroup of 𝐺 containing G ⁢ ( ω ) . Equivalently, ∼ is the coarsest nonuniversal 𝐺-invariant equivalence relation. In particular, we can recover ∼ from the action of 𝐺 on Ω.

Proof

Since 𝐺 is 𝑘-by-block-transitive, certainly 𝐺 is transitive and there are k ≥ 2 blocks, so ∼ cannot be the universal relation. We see that ∼ is 𝐺-invariant because if it were not, say x ∼ y but g ⁢ x ≁ g ⁢ y , then we would have a distant 𝑘-tuple ( g ⁢ x , g ⁢ y , x 3 , … , x k ) in the same 𝐺-orbit as the nondistant 𝑘-tuple

In particular, we have G ⁢ ( ω ) ≤ G ⁢ ( [ ω ] ) < G .

Suppose now that 𝐻 is some subgroup of 𝐺 containing G ⁢ ( ω ) , and suppose that H ≰ G ⁢ ( [ ω ] ) , say g ∈ H ∖ G ⁢ ( [ ω ] ) ; let 𝑂 be the 𝐻-orbit of g ⁢ ω . Then g ⁢ ω ∈ Ω ∖ [ ω ] ; since H ≥ G ⁢ ( ω ) and G ⁢ ( ω ) acts transitively on Ω ∖ [ ω ] , it follows that Ω ∖ [ ω ] ⊆ O ; in particular, [ g ⁢ ω ] ⊆ O . But now also g − 1 ⁢ [ g ⁢ ω ] = [ ω ] ⊆ O , so in fact, O = Ω , that is, 𝐻 is transitive on Ω. Since G ⁢ ( ω ) ≤ H , we conclude that H = G .

From the previous paragraph, we deduce that ∼ is the coarsest 𝐺-invariant equivalence relation other than the universal relation. ∎

Let Ω be a set, let G ≤ Sym ⁢ ( Ω ) be block-faithful and 𝑘-by-block-transitive on Ω relative to the nontrivial equivalence relation ∼ , for k ≥ 2 . Let 𝑁 be a nontrivial normal subgroup of 𝐺. Then G = N ⁢ G ⁢ ( ω ) . In particular, 𝑁 is not abelian.

Proof

Since 𝐺 acts 𝑘-transitively on the set of blocks, we see that 𝑁 acts transitively on the set of blocks, so N ≰ G ⁢ ( [ ω ] ) . It then follows by Lemma 2.3 that G = N ⁢ G ⁢ ( ω ) . Since ∼ is nontrivial, we have G ⁢ ( ω ) < G ⁢ ( [ ω ] ) , so N ∩ G ⁢ ( [ ω ] ) is nontrivial. Since 𝑁 acts faithfully on the set of blocks, we deduce that 𝑁 is not abelian. ∎

Let 𝐺 be a group and let L < K < G be proper subgroups. If the action of 𝐺 on G / L is 2-by-block-transitive with block stabilizer M ≥ L , then M ≥ K and the action on G / K is 2-by-block-transitive with block stabilizer 𝑀.

Proof

Suppose 𝐺 has 2-by-block-transitive action on Ω = G / L and write 𝜔 for the trivial coset. Then M = G ⁢ ( [ ω ] ) ; by Lemma 2.3, since L < K < G , we have K ≤ M . Let g ∈ G ∖ M . By Lemma 2.1, we have G = L ⁢ g ⁢ L ∪ M ; it then follows that G = K ⁢ g ⁢ K ∪ M , so the action of 𝐺 on G / K is 2-by-block-transitive by Lemma 2.1. ∎

Let Ω be a set and let 𝐺 be a group acting on Ω that preserves the nonuniversal equivalence relation ∼ , acts faithfully and 2-transitively on Ω / ∼ and acts transitively on Ω.

1. Given ω , ω ′ ∈ Ω , ω ≁ ω ′ , there is an involution s ∈ G such that s ⁢ [ ω ] = [ ω ′ ] and such that

   G ⁢ ( { [ ω ] , [ ω ′ ] } ) = G ⁢ ( [ ω ] , [ ω ′ ] ) ⋊ ⟨ s ⟩ .

2. Let ω ∈ Ω , let L = G ⁢ ( ω ) and let s ∈ G be an involution such that s ⁢ ω ≁ ω . Then 𝐺 has 2-by-block-transitive action on Ω if and only if

   (2.1) G ⁢ ( [ ω ] ) = L ⁢ s ⁢ L ⁢ ( [ s ⁢ ω ] ) ⁢ s .

   In particular, if 𝐺 has 2-by-block-transitive action on Ω, then

   (2.2) G ⁢ ( [ ω ] , [ s ⁢ ω ] ) = L ⁢ ( [ s ⁢ ω ] ) ⁢ s ⁢ L ⁢ ( [ s ⁢ ω ] ) ⁢ s ;

   if in addition 𝐺 is finite, then

   (2.3) | G ( [ ω ] , [ s ω ] ) : L ( [ s ω ] ) | = | G ( [ ω ] ) : L | .

Proof

(i) Since 𝐺 acts 2-transitively on Ω / ∼ , there is an element swapping two blocks, so 𝐺 has an element of even order, and hence an element 𝑠 of order 2. Since 𝐺 acts faithfully on Ω / ∼ , there is some block not fixed by 𝑠, say s ⁢ [ ω 0 ] = s ⁢ [ ω 0 ′ ] where ω 0 , ω 0 ′ ∈ Ω such that ω 0 ≁ ω 0 ′ . Since 𝐺 is 2-transitive on Ω / ∼ , after conjugating 𝑠 by a suitable element of 𝐺, we can in fact take [ ω 0 ] = [ ω ] and [ ω 0 ′ ] = [ ω ′ ] . The form taken by G ⁢ ( { [ ω ] , [ ω ′ ] } ) is now clear.

(ii) We have G ⁢ ( [ ω ] ) = L ⁢ s ⁢ L ⁢ ( [ s ⁢ ω ] ) ⁢ s if and only if 𝐿 acts transitively on the coset space X = G ⁢ ( [ ω ] ) / s ⁢ L ⁢ ( [ s ⁢ ω ] ) ⁢ s . As a G ⁢ ( [ ω ] ) -space, 𝑋 is equivalent to the G ⁢ ( [ ω ] ) -orbit 𝑌 of s ⁢ ω since

Since 𝐺 acts 2-transitively on blocks, we see that 𝑌 intersects every block other than [ ω ] .

If 𝐺 is 2-by-block-transitive, then we have Y = Ω ∖ [ ω ] and 𝐿 acts transitively on Ω ∖ [ ω ] , so we deduce (2.1) from the previous paragraph, and then (2.2) follows from (2.1) by intersecting both sides with the group G ⁢ ( [ s ⁢ ω ] ) . If 𝐺 is finite, then (2.3) follows from (2.2) and the fact that 𝐿 acts transitively on ( Ω / ∼ ) ∖ { [ ω ] } so that | L : L ( [ s ω ] ) | = | G ( [ ω ] ) : G ( [ ω ] , [ s ω ] ) | .

Conversely, assume (2.1). Then 𝐿 acts transitively on 𝑌, so 𝐿 acts transitively on ( Ω / ∼ ) ∖ { [ ω ] } . We can write the last statement as a double coset equation, using the fact that [ s ⁢ ω ] ≠ [ ω ] , as follows:

equivalently,

which means that G ⁢ ( [ ω ] ) has only two orbits on Ω, one of which is Ω ∖ [ ω ] . Thus, by (2.1), in fact, 𝐿 acts transitively on Ω ∖ [ ω ] ; hence 𝐺 acts transitively on distant pairs. ∎

Let Ω be a set with at least 3 elements, let 𝐺 be a group acting 2-transitively on Ω and let L ≤ G ⁢ ( ω ) be such that 𝐺 acts 2-by-block-transitively on G / L . Then, for all g ∈ G , we have g ⁢ L ⁢ g − 1 ≤ G ⁢ ( ω ) if and only if g ∈ G ⁢ ( ω ) .

Proof

If g ∈ G ⁢ ( ω ) , then clearly g ⁢ L ⁢ g − 1 ≤ G ⁢ ( ω ) .

Conversely, suppose g ∈ G is such that g ⁢ L ⁢ g − 1 ≤ G ⁢ ( ω ) . Then, by Lemma 2.3, g ⁢ G ⁢ ( ω ) ⁢ g − 1 is the largest proper subgroup of 𝐺 that contains g ⁢ L ⁢ g − 1 , so

Since 𝐺 acts 2-transitively on Ω, the subgroup G ⁢ ( ω ) is already maximal among proper subgroups, so we must have G ⁢ ( ω ) = g ⁢ G ⁢ ( ω ) ⁢ g − 1 , that is, g ∈ N G ⁢ ( G ⁢ ( ω ) ) . There are two possibilities for the normalizer: either

In the former case, we are done. In the latter case, G ⁢ ( ω ) fixes Ω pointwise, which is incompatible with 2-transitivity given the assumption that | Ω | ≥ 3 . ∎

Based on Lemma 2.1, one can classify the 2-by-block-transitive actions of a given finite group 𝐺 by the following simple (but not particularly efficient) procedure.

1. Determine a set ℳ of representatives of the conjugacy classes of subgroups 𝐻 of 𝐺 such that 𝐺 acts 2-transitively on G / H .

2. For each H ∈ M , fix g H ∈ G ∖ H .

3. Produce a sequence L i H of sets of subgroups of 𝐻, starting with L 0 H = { H } . For each L ∈ L i H , we take representatives of the 𝐻-conjugacy classes occurring as maximal subgroups 𝑀 of 𝐿; then for each representative 𝑀, we check if G = M ⁢ g H ⁢ M ∪ H (e.g. by calculating whether | M ⁢ g H ⁢ M | + | H | = | G | ). We then form a set L i + 1 H consisting of those 𝑀 in the previous sentence such that G = M ⁢ g H ⁢ M ∪ H .

4. The previous step eventually terminates; write L H = ⋃ i ≥ 0 L i H .

5. Write L = ⋃ H ∈ M L H . Up to equivalence, the 2-by-block-transitive actions of 𝐺 are given by the action on G / L for L ∈ L .

Lemma 2.9 (Hering [6]; see also [10, Appendix 1])

Let H = V ⋊ G be a finite 2-transitive affine permutation group, where 𝑉 is regarded as the additive group of some vector space on which 𝐺 acts by semilinear maps. Then 𝐺 and 𝑉 can be taken as follows:

1. 𝑉 is the field of order 𝑞, and G ≤ Γ ⁢ L 1 ⁢ ( q ) ;

2. 𝑉 is the vector space of dimension 𝑛 over the field of order 𝑞, and we have SL n ⁢ ( q ) ⊴ G ;

3. 𝑉 is the vector space of dimension 2 ⁢ m over the field of order 𝑞, and we have Sp 2 ⁢ m ⁢ ( q ) ⊴ G ;

4. 𝑉 is the vector space of dimension 6 over the field of order q = 2 e , and G 2 ⁢ ( q ) ⊴ G ;

5. 𝐻 is one of finitely many exceptional 2-transitive groups of affine type, with 𝑉 being a vector space of dimension 2, 4 or 6. In each case, either 𝐻 is soluble or 𝐻 has a unique nonabelian composition factor, which is one of Alt ⁢ ( 5 ) , Alt ⁢ ( 6 ) , Alt ⁢ ( 7 ) , PSL 2 ⁢ ( 13 ) . If 𝑉 has dimension 2, then

   q ∈ { 5 , 7 , 9 , 11 , 19 , 23 , 29 , 59 } .

Let 𝐺 be one of

acting 2-transitively on a set 𝑋 of 𝑑 points, where d = 11 , 15 , 28 , 176 , 276 respectively. Then 𝑋 does not extend properly to a 2-by-block-transitive action of 𝐺.

Proof

Let 𝑥 and 𝑦 be distinct elements of 𝑋 and let 𝑠 be an involution of 𝐺 such that s ⁢ x = y and s ⁢ y = x , as in Lemma 2.6 (i). Let L ≤ G ⁢ ( x ) be a point stabilizer of a 2-by-block-transitive action of 𝐺.

For G = PSL 2 ⁢ ( 11 ) acting on 11 points, we see that | G ⁢ ( x , y ) | = 6 , and for G = P ⁢ Γ ⁢ L 2 ⁢ ( 8 ) acting on 28 points, | G ⁢ ( x , y ) | = 2 . For G = HS acting on 176 points, we have G ⁢ ( x ) = PSU 3 ⁢ ( 5 ) ⋊ C 2 and G ⁢ ( x , y ) = Aut ⁢ ( Alt ⁢ ( 6 ) ) , so every automorphism of G ⁢ ( x , y ) is inner. In all of these cases, we see that there is no proper subgroup 𝐻 of G ⁢ ( x , y ) such that G ⁢ ( x , y ) = H ⁢ s ⁢ H ⁢ s . The conclusion now follows by Lemma 2.6.

For G = Alt ⁢ ( 7 ) acting on 15 points, we have | G | − | G ⁢ ( x ) | = 2 4 ⋅ 3 ⋅ 7 2 , so by Corollary 2.2, we would need | L | to be a multiple of 2 2 ⋅ 3 ⋅ 7 , or in other words, index at most 2 in G ⁢ ( x ) . However, G ⁢ ( x ) ≅ PSL 3 ⁢ ( 2 ) is simple and therefore has no subgroup of index 2. Thus there is no proper extension of 𝑋 to a 2-by-block-transitive action in this case.