ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.09
https://doi.org/10.26493/1855-3974.2751.81f
(Also available at http://amc-journal.eu)
Finite simple groups on triple systems*
Xiaoqin Zhan , Xuan Pang , Suyun Ding †
School of Science, East China JiaoTong University,
Nanchang, 330013, People’s Republic of China
Received 3 December 2021, accepted 14 May 2023, published online 22 November 2023
Abstract
Let D be a triple system, and let G be a finite simple group. In this paper we almost
determine all possibilities of D admitting G as its flag-transitive automorphism group.
Keywords: Triple system, flag-transitivity, finite simple group.
Math. Subj. Class. (2020): 05B07, 20B25, 05B25
1 Introduction
A 2-(v, k, λ) design is a pair D = (P,B) where P is a set of v points and B is a collection
of b k-subsets (blocks) of P with the property that every 2-subset of P occurs in λ blocks
of B. If no blocks are identical, then D is called simple.
An automorphism of a design D is a permutation of P which leaves B invariant. The
full automorphism group of D, denoted by Aut(D), is the group consisting of all automor-
phisms of D. A flag of D is a point-block pair (α,B) such that α ∈ B. For G ≤ Aut(D),
G or D is called flag-transitive if G acts transitively on the set of flags, and point-primitive
if G acts primitively on P . A set of blocks of D is called a set of base blocks with respect
to an automorphism group G of D if it contains exactly one block from each G-orbit on the
block set. In particular, if G is a flag-transitive automorphism group of D, then any block
B is a base block of D.
*The authors would like to express their gratitude to the referee who made very helpful comments and sugges-
tions that improved our paper. This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12361004 and 11961026) and the Natural Science Foundation of Jiangxi Province (Grant Nos.
20224BAB211005 and 20224BAB201005).
†Corresponding author.
E-mail addresses: zhanxiaoqinshuai@126.com (Xiaoqin Zhan), p1443202623@163.com (Xuan Pang),
dingsy2017@163.com (Suyun Ding)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P2.09
In this paper, we focus on simple 2-(v, 3, λ) designs also known as simple triple sys-
tems, which can be denoted by TS(v, λ). One possibility is to take all possible 3-subsets
of P however such designs are called complete and will be ignored. A triple system is a
Steiner triple system, or STS(v), when λ = 1.
Let r be the number of the blocks through a given point. For a TS(v, λ), it is well
known that a necessary and sufficient condition for the existence of a TS(v, λ) is v ̸= 2
and λ ≡ 0 (mod (v − 2, 6)), and
3b = vr; (1.1)
r =
λ(v − 1)
2
; (1.2)
b =
λv(v − 1)
6
; (1.3)
b ≥ v. (1.4)
A 2-(v, k, 1) design is also called a finite linear space. A classic result is that of Higman
and McLaughlin [8] who proved that for a finite linear space, flag-transitivity implies point-
primitivity. Then Buekenhout, Delandtsheer and Doyen in [1] proved that if G acts flag-
transitively on a linear space, then G is of affine or almost simple type. In 1990, the
six-person team [2] classified all flag-transitive linear spaces apart from those with an one-
dimensional affine automorphism group.
For 2-(v, k, 1) designs with small values of k, one of the first classifications was for
Steiner triple systems in [4], which considered what happens when the action was block-
transitive but not 2-transitive on points. It is described in [11] what happens when the
action on points is 2-transitive. This result depends on the classification of all finite simple
groups and is subsumed into the general results proved by Kantor in [10].
Let G be a flag-transitive automorphism group of a TS(v, λ). It is shown in [6, 2.3.7(c),
(e)] that G is point-primitive. Moreover, we can easily prove that G is 2-homogeneous (see
Lemma 2.2 below). This result makes it possible to classify all flag-transitive triple systems
using the classification of the finite 2-transitive permutation groups. Our main purpose is
to give a classification of all triple systems admitting a simple flag-transitive automorphism
group.
We now state the main result of this paper:
Theorem 1.1. Let D be a triple system, and let G be a finite simple group. If G acts
flag-transitively on D, then one of the following LINES of Table 1 holds.
Remark 1.2.
• All but the triple systems listed in LINES 20 and 21 exist.
• If G = PSU(3, q) with q = 5, then there are only two flag-transitive triple systems
corresponding to LINES 19 and 20.
• The existence of triple systems with 3 ∤ q and q ̸= 5 corresponding to LINES 20 and
21 is in doubt.
X. Zhan et al.: Finite simple groups on triple systems 3
Table 1: G and corresponding triple systems.
LINE G D Notes
1 A7 TS(15, 1)
2 TS(15, 12)
3 PSL(2, 11) TS(11, 3)
4 TS(11, 6)
5 HS TS(176, 12)
6 TS(176, 72)
7 TS(176, 90)
8 Co3 TS(276, 112)
9 TS(276, 162)
10 PSp(2d, 2) TS(2d−1(2d + 1), 22d−2) d ≥ 3
11 TS(2d−1(2d + 1), 2(2d−1 − 1)(2d−2 + 1))
12 PSp(2d, 2) TS(2d−1(2d − 1), 22d−2) d ≥ 3
13 TS(2d−1(2d − 1), 2(2d−1 + 1)(2d−2 − 1))
14 PSL(d, q) TS( q
d−1
q−1 , q − 1) d ≥ 3
15 TS( q
d−1
q−1 ,
qd−1
q−1 − q − 1)
16 PSL(2, q) TS(q + 1, q−1
2
) q ≡ 1(mod 4)
17 Ree(q) TS(q3 + 1, 2(q − 1)) q = 32e+1 > 3
18 TS(q3 + 1, q − 1)
19 PSU(3, q) TS(q3 + 1, q − 1) q ≥ 3
20 TS(q3 + 1, q
2−1
(3,q+1)
)
21 TS(q3 + 1, 2(q
2−1)
(3,q+1)
)
2 Useful lemmas
The notation and terminology used is standard and can be found in [5, 6] for design theory
and in [7, 9] for group theory. In particular, if G is a permutation group on a set Ω, and
{α, β} ⊆ ∆ ⊆ Ω, then Gα denotes the stabilizer of a point α in G, and Gαβ denotes the
pointwise stabilizer of two points α and β in G, and G∆ denotes the setwise stabilizer of
∆ in G.
The following result about flag-transitive 2-designs is well-known.
Lemma 2.1. Let D = (P,B) be a 2-(v, k, λ) design, and let G be an automorphism group
of D. For any α ∈ P and B ∈ B, G is flag-transitive if and only if G is point-transitive
and Gα is transitive on the pencil P (α) (the set of blocks through α), if and only if G is
block-transitive and GB is transitive on the points of B.
Lemma 2.2. Let D = (P,B) be a triple system, and let G be a flag-transitive automor-
phism group of D. If G is a simple group, then G acts 2-transitively on P .
Proof. Let {α, β} and {γ, δ} be arbitrary two unordered pairs of P . By the definition of a
triple system, there are two points ε and θ such that B1 = {α, β, ε} and B2 = {γ, δ, θ} are
two blocks of D. The flag-transitivity of G implies that there is a g ∈ G such that
(ε,B1)
g = (εg, Bg1 ) = (θ,B2),
and so {α, β}g = {γ, δ}. Thus G is 2-homogeneous. If G is a simple group, then G acts
2-transitively on P by [7, Theorem 9.4B].
4 Ars Math. Contemp. 24 (2024) #P2.09
Lemma 2.3. Let D = (P,B) be a triple system, and let G ≤ Aut(D) be a 2-transitive
group on P . Then the following conditions are equivalent:
(i) G acts flag-transitively on D.
(ii) If B = {α, β, γ} ∈ B, then {{α, β, γi} | γi ∈ γG{α,β}} is the set of all blocks
through points α and β.
Proof. (i) ⇒ (ii): Let B(α, β) = {B1, B2, . . . , Bλ} be the set of blocks through points α
and β, where Bi = {α, β, γi}, γi ∈ P \ {α, β}. Clearly, B(α, β)G{α,β} = B(α, β). If G
acts flag-transitively on D, then for any two flags (γi, Bi) and (γj , Bj), there is a g ∈ G
such that (γi, Bi)g = (γj , Bj), so γ
g
i = γj and g ∈ G{α,β}. Thus G{α,β} acts transitively
on B(α, β) and hence {γ1, . . . , γλ} = γ
G{α,β}
i .
(ii) ⇒ (i): Let (γ,B) and (ϵ, C) be two flags of D with B = {α, β, γ}, C = {δ, η, ϵ}.
By the 2-transitivity of G, there exists g1 ∈ G such that {α, β}g1 = {δ, η}, thus Bg1 =
{δ, η, γg1} is a block containing δ and η. Since {{δ, η, ϵi} | ϵi ∈ ϵG{δ,η}} is the set
of all blocks through δ and η, there exists g2 ∈ G{δ,η} such that γg1g2 = ϵ, and then
(γ,B)g1g2 = (ϵ, C). Therefore, G acts flag-transitively on D.
Corollary 2.4. Let G be a 2-transitive group on a point set P with |P| = v, and let
λ1, λ2, . . . , λk be all sizes of orbits of Gαβ on P \ {α, β}. If λi ̸= λj for i ̸= j, then there
exist k different flag-transitive TS(v, λi).
Proof. Without loss of generality, let ∆ = γGαβ with |∆| = λ1, where γ ∈ P \ {α, β}.
Since Gαβ ⊴ G{α,β}, the group Gαβ acts 12 -transitively on γ
G{α,β} , that is, Gαβ-orbits
on γG{α,β} have the same length. The uniqueness of the Gαβ-orbit with size λ1 implies
that γGαβ = γG{α,β} . Thus G{α,β} has a unique orbit with size λ1. Let B = {α, β, γ}
and B = BG. We shall prove below that D = (P,B) is a TS(v, λ1) admitting G as its
flag-transitive automorphism group.
Since G is 2-transitive, for any pair {δ, η}, there exists g ∈ G such that {α, β}g =
{δ, η}. So G{δ,η} has a unique orbit ∆g = (γg)G{δ,η} with |∆g| = |∆| = λ1. Let
B(δ, η) be the set of elements of B containing δ, η with |B(δ, η)| = λ. It is easy to see that
Λ = {{δ, η, ϵ} | ϵ ∈ ∆g} ⊆ B, so we have λ ≥ λ1. On the other hand, for C = {δ, η, θ} ∈
B(δ, η), there exists h ∈ G such that C = Bh. As |γGαβ | = |αGγβ | = |βGαγ | = λ1,
we may assume that θ = γh. Then |θG{δ,η} | = |γhG{δ,η} | = |γG{α,β}h| = |∆h| = λ1,
it implies λ1 ≥ λ. Thus, λ = λ1 and B(δ, η) = Λ. Hence D is a TS(v, λ1), and G is a
flag-transitive automorphism group of D by Lemma 2.3(ii).
Lemma 2.5. Let G be a 2-transitive group on a point set P with |P| = v, and let ∆ =
{α, β, γ} be a 3-subset of P . If Gαβ is a cyclic group of order λ and |γGαβ | = λ, then
(i) D = (P,∆G) is a flag-transitive TS(v, λ) if and only if G∆∆ ∼= S3, or
(ii) D = (P,∆G) is a flag-transitive TS(v, 2λ) if and only if G∆∆ ∼= Z3.
Proof. Here we only prove case (i), and case (ii) can be proved by same procedure. Since
Gαβ is a cyclic group for any points α and β, we have that G∆ = G∆∆. Let D = (P,∆G).
If D is a flag-transitive TS(v, λ), then using Lemma 2.1 and Equation (1.3), we have that
b =
λv(v − 1)
6
= |∆G| = [G : G∆] = [G : Gαβ ][Gαβ : G∆].
X. Zhan et al.: Finite simple groups on triple systems 5
By 2-transitivity of G and |Gαβ | = λ, we obtain |G∆| = 6. The flag-transitivity of G
implies that G∆ acts transitively on the points of ∆ by Lemma 2.1. Thus G∆ ∼= S3.
If G∆ ∼= S3, then G{α,β}γ ∼= Z2 and |∆G| = [G : Gαβ ][Gαβ : G∆] = λv(v−1)6 . Thus,
D is a TS(v, λ) as G acts 2-transitively on P . Clearly,
|γG{α,β} | = [G{α,β} : G{α,β}γ ] = λ,
where G{α,β}γ = G{α,β} ∩Gγ . Therefore, G acts flag-transitively on D by Corollary 2.4.
Lemma 2.6. Let G = Ree(q) act 2-transitively on Ω, where |Ω| = q3+1 and q = 32e+1 >
3. Then there exist subsets ∆, Σ of size 3 such that
G∆∆ = Z3, GΣΣ = S3.
Proof. Let Q be a Sylow 3-subgroup of G. Then |Q| = q3, and there exists α ∈ Ω such that
Q is regular on Ω\{α}. Thus each subgroup of Q is semiregular on Ω\{α}. Let x, y ∈ Q
such |x| = |y| = 3, x /∈ Z(Q) and y ∈ Z(Q), where the centre Z(Q) is elementary abelian
of order q.
Let ∆ be an orbit of ⟨x⟩. Then |∆| = 3 and G∆∆ = Z3 or S3. Further, since x is not
conjugate to x−1 in G (reference [12]), we have G∆∆ ∼= ⟨x⟩ ∼= Z3.
Consider y acting on Ω \ {α}. Since y is in the centre Z(Q), there is an involution
z ∈ Gα such that yz = y−1, and the subgroup H = ⟨y, z⟩ ∼= S3. Since ⟨y⟩ is semiregular
on Ω \ {α}, the set Ω \ {α} is divided into 13q
3 orbits of ⟨y⟩:
∆1,∆2, . . . ,∆m,
where m = 13q
3 is odd. Since each H-orbit Σ contains a ⟨y⟩-orbit, the cardinality |Σ| = 3
or 6. As the number 13q
3 of ⟨y⟩-orbits is odd, it follows that there is at least one H-orbit Σ
on Ω \ {α} has length 3. Therefore, GΣΣ = HΣΣ = S3 with |Σ| = 3.
3 Proof of Theorem 1.1
Let D = (P,B) be a TS(v, λ), and let G be a simple group acting flag-transitively on D.
Then G acts 2-transitively on P by Lemma 2.2. Since we neglect the case D is complete,
we may assume that G is not 3-homogeneous group on P . Thus, all such groups are known
and we can find a classification in [3] and we have that G must be one of the following
Table 2.
We will prove Theorem 1.1 by analyzing the 11 cases in Table 2 one by one.
Proof of Theorem 1.1. Let α and β be two points of P . For Cases 1 – 7, we have the
following facts by the proof of [10, Theorem 1]:
If G = A7 and v = 15, then Gαβ has orbit-lengths 1 and 12 on P \ {α, β}.
If G = PSL(2, 11) and v = 11, then Gαβ has orbit-lengths 3 and 6 on P \ {α, β}.
If G = HS and v = 176, then Gαβ has orbit-lengths 12, 72 and 90 on P \ {α, β}.
If G = Co3 and v = 276, then Gαβ has orbit-lengths 112 and 162 on P \ {α, β}.
If G = PSp(2d, 2) and v = 22d−1 + 2d−1, then Gαβ has orbit-lengths 2(2d−1 −
1)(2d−2 + 1) and 22d−2 on P \ {α, β}.
6 Ars Math. Contemp. 24 (2024) #P2.09
Table 2: 2-transitive, not 3-homogeneous simple groups.
Case Group Degree Notes
1 A7 15
2 PSL(2, 11) 11
3 HS 176
4 Co3 276
5 PSp(2d, 2) 22d−1 + 2d−1 d ≥ 3
6 PSp(2d, 2) 22d−1 − 2d−1 d ≥ 3
7 PSL(d, q) (qd − 1)/(q − 1) d ≥ 3
8 PSL(2, q) q + 1 q ≡ 1 (mod 4)
9 Suz(q) q2 + 1 q = 22e+1 > 2
10 Ree(q) q3 + 1 q = 32e+1 > 3
11 PSU(3, q) q3 + 1 q ≥ 3
If G = PSp(2d, 2) and v = 22d−1 − 2d−1, then Gαβ has orbit-lengths 2(2d−1 +
1)(2d−2 − 1) and 22d−2 on P \ {α, β}.
If G = PSL(d, q) with d ≥ 3 and v = q
d−1
q−1 , Gαβ has orbit-lengths q − 1 and
qd−1
q−1 − q − 1 on P \ {α, β}.
It follows from Corollary 2.4 that D is one of triple systems corresponding LINES 1-15
in Table 1.
Case 8: G = PSL(2, q) with q ≡ 1 (mod 4) and v = q + 1. In this case, there are
exactly two G-orbits on 3-subsets of q + 1 points with size q(q
2−1)
12 . Also, Gαβ
∼= Z q−1
2
has two orbits with length q−12 on P \ {α, β}, denoted by Γ1 and Γ2. Suppose that Γ1 =
{α1, α2, . . . , α q−1
2
}, Γ2 = {β1, β2, . . . , β q−1
2
}. For i ∈ {1, 2}, let Di = (P,∆Gi ) where
∆i = {α, β, γi} and γi ∈ Γi. It is easy to calculate that |G∆i | = 6, and hence G∆i ∼= S3.
By Lemma 2.5(i), both D1 and D2 are TS(q + 1, q−12 ). Let
g = (α, β)(α1, β1) · · · (α q−1
2
, β q−1
2
).
Clearly, g is an isomorphism from D1 to D2, that is D1 ∼= D2. Thus, D is a TS(q+1, q−12 ).
Case 9: G = Sz(q) and v = q2 + 1. Since G acts flag-transitively on D, then 3 | |G| by
Lemma 2.1. But this contracts the fact that 3 ∤ |G| (see [9, Theorem 3.6]). Therefore, there
is no triple system admitting Sz(q) as its flag-transitive automorphism group.
Case 10: G = Ree(q) and v = q3 +1 with q = 32e+1 > 3. From Lemmas 2.5 and 2.6, we
have that D is one of triple systems corresponding LINES 17 and 18 in Table 1.
Case 11: G = PSU(3, q) and v = q3 + 1. Since Gαβ ∼= Z q2−1
(3,q+1)
has a unique orbit O
with size q − 1 and q(3, q + 1) orbits with size q
2−1
(3,q+1) . Similar to proof of Lemma 2.4,
we can prove that there exists a unique TS(q3 +1, q− 1) admitting G as its flag-transitive
automorphism group.
If q = 3e ≥ 3, there exist subsets ∆, Σ of size 3 such that G∆∆ = Z3, GΣΣ = S3 by the
same proof as Lemma 2.6. In this case, D is one of triple systems corresponding LINES 20
and 21 in Table 1 from Lemma 2.5.
X. Zhan et al.: Finite simple groups on triple systems 7
If q = 5 then D can only be a flag-transitive TS(126, 8) in addition to TS(126, 4) by a
simple calculation. This means that there is no flag-transitive TS(126, 16) in this case.
Unfortunately, we don’t know whether Lemma 2.6 holds when 3 ∤ q. Thus the existence
of TS(q3 + 1, q
2−1
(3,q+1) ) (or TS(q
3 + 1, 2(q
2−1)
(3,q+1) )) with 3 ∤ q and q ̸= 5 is in doubt.
This completes the proof of Theorem 1.1.
Conjecture 3.1. Let D be a triple system TS(q3 + 1, λ), and let G = PSU(3, q) act
flag-transitively on D with 3 ∤ q and q ̸= 5. If λ ̸= q − 1 then one of following holds:
(i) If q is even, then λ = 2(q
2−1)
(3,q+1) .
(ii) If q is odd, then λ = q
2−1
(3,q+1) or
2(q2−1)
(3,q+1) .
In fact, using MAGMA, we have already proved that the conjecture holds when q ≤ 100.
ORCID iDs
Xiaoqin Zhan https://orcid.org/0000-0003-0669-6419
Xuan Pang https://orcid.org/0000-0003-2500-9741
Suyun Ding https://orcid.org/0000-0002-6564-4427
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