Flag-transitive $2$-designs with block size 5 and alternating groups

1.   Introduction

A 2-$(v, k, \lambda)$ design (or a 2-design in short) is a structure $(\mathcal{P}, \mathcal{B})$ consisting of a set $\mathcal{P}$ of $v$ elements (called points) and a set $\mathcal{B}$ of some $k$-subsets (called blocks) of $\mathcal{P}$, satisfying the condition that any two points of $\mathcal{P}$ are contained in exactly $\lambda$ blocks in $\mathcal{B}$. The number of blocks containing a fixed point is proved to be a constant and denoted by the letter $r$. The total number $|{\mathcal{B}}|$ of blocks is denoted by the letter $b$. These $v$, $k$, $\lambda$, $b$ and $r$ are called the parameters of a design. If

then $\mathcal{D}$ is called a complete design. We only consider incomplete designs in this paper. We shall also denote a 2-design by $\mathcal{D}$. A point-block pair $(x, B)$ such that $x\in B$ is called a flag of a 2-design. We denote by $\mathcal{F}$ the set of all flags in a 2-design.

A permutation $g$ of the point set $\mathcal{P}$ of a 2-design

is an automorphism if $g$ preserves the block set $\mathcal{B}$. All automorphisms of $\mathcal{D}$ form a group with the natural product of permutations, which is defined as the full automorphism group of ${\mathcal{D}}$. We usually denoted this group by ${{{\rm{Aut}}}}(\mathcal{D})$. A group $G$ is called an automorphism group of $\mathcal{D}$ if $G$ is a subgroup of ${{{\rm{Aut}}}}(\mathcal{D})$. An automorphism group of a 2-design ${\mathcal{D}}$ is called flag-transitive on ${\mathcal{D}}$ if it acts transitively on $\mathcal{F}$. Similarly, a group is point-primitive if it is primitive on the point set $\mathcal{P}$, and one can define other types of transitive actions such as block-transitive and point-transitive.

In the present paper we shall adopt usual notations in finite permutation groups, which shall be consistent with those in ^[1] or ^[2], for example. Let $G$ be an automorphism group of a 2-design

Commonly, we write $G_x$ as the stabilizer of the point $x\in{{\mathcal P}}$ in $G$ and write $G_{B}$ as the block-stabilizer of the block $B\in{{\mathcal B}}$ in $G$. We always denote by $\Omega_n$ a set of cardinality $n$. Moreover, we denote by ${{{\rm{Alt}}}}(\Omega_n)$ (respectively, ${{{\rm{Sym}}}}(\Omega_n)$) the alternating group (respectively, the symmetric group) on $\Omega_n$. Sometimes, in abbreviation, we also adopt the notation ${{{\rm{Alt}}}}(n)$ (or even shorter, ${\rm A_n}$) and ${{{\rm{Sym}}}}(n)$ (or ${\rm S_n}$).

Classifying flag-transitive 2-designs is a long-term project. A very classic result on flag-transitive finite linear spaces, namely, $2$-$(v, k, 1)$ designs, is given by Kantor ^[3] and a team of six ^[4]. Some classifications for symmetric designs with special automorphism groups are studied in ^[5,6,7], for 2-transitive groups and primitive groups of rank 3, respectively. Inspecting flag-transitive 2-designs with their block size a small number is also an important research direction. A 2-design with block size 2 has trivial structure, having the block set all the 2-subsets of points, and a block-transitive automorphism group of such a design is 2-homogeneous. Reductions for flag-transitive 2-designs with block sizes 3 and 4 are tackled in ^[8]. Then, flag-transitive 2-designs with block size 5 were studied by the first author and Zhou in ^[9]. They reduced such flag-transitive automorphism groups to point-primitive of affine type and almost simple type and gave a classification on the case that the groups have a sporadic simple socle.

The well-known Classification of Finite Simple Groups states that a finite non-abelian simple group is isomorphic to one of the groups in the following four types: (1) alternating groups; (2) classical simple groups; (3) exceptional simple groups of Lie type; and (4) sporadic simple groups. Since the authors in ^[9] have dealt with flag-transitive 2-designs with block size 5 in terms of sporadic simple groups. As a continuation of this classification project, in the current paper we will classify such 2-designs admitting a flag-transitive almost simple automorphism group whose socle is a simple alternating group. A complete classification is given in the following Theorem 1.1, as the main result of the paper.

Theorem 1.1. Let $\mathcal{D}$ be a $2$-$(v, 5, \lambda)$ design, and let $G$ be an almost simple, flag-transitive, point-primitive automorphism group of $\mathcal{D}$ with alternating socle ${{{\rm{Alt}}}}(n)$ ($n\geq5$). Then one of the following holds:

(1) ${\mathcal{D}}$ is a unique $2$-$(10, 5, 8)$ design with $G = {{{\rm{Alt}}}}(6)$ or ${{{\rm{M}}}}_{10}$.

(2) ${\mathcal{D}}$ is a unique $2$-$(10, 5, 16)$ design with $G = {{{\rm{Sym}}}}(6)$, ${{{\rm{PGL}}}}_2(9)$, or ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$.

(3) ${\mathcal{D}}$ is a unique $2$-$(21, 5, 12)$ design with $G = {{{\rm{Alt}}}}(7)$ or ${{{\rm{Sym}}}}(7)$.

(4) ${\mathcal{D}}$ is a unique $2$-$(15, 5, 4)$ design with $G = {{{\rm{Alt}}}}(7)$.

(5) ${\mathcal{D}}$ is a unique $2$-$(15, 5, 12)$ design with $G = {{{\rm{Alt}}}}(7)$.

(6) ${\mathcal{D}}$ is a unique $2$-$(15, 5, 16)$ design with $G = {{{\rm{Alt}}}}(8)$.

Further, all the groups $G$ are 2-transitive on $\mathcal{P}$, except the group $G$ in (3), which acts on $\mathcal{P}$ with rank 3 and with the two suborbits of length 10.

Remark 1.1 It is an important fact that for a fixed parameter set $(v, k, \lambda)$, there could be many non-isomorphic designs with such parameters. In fact, for Theorem 1.1 (1), (2), and (4), check [10, Section Ⅱ. 1.3] and we can see that there are more than 135922,108, and 896 non-isomorphic designs with

respectively. However, under the condition that the design admits a flag-transitive automorphism group with the socle a simple alternating group, there is only one design with each above parameter set.

Combining the above main theorem with the results in ^[9], we have the following immediate corollary.

Corollary 1. If $G$ is a flag-transitive automorphism group of a $2$-$(v, 5, \lambda)$ design. Then $G$ is point-primitive of affine type or almost simple type with socle a simple group of Lie type, or is one of the cases listed in Theorem 1.1 and [9, Theorem 2].

According to the result, we see that the remaining work of classifying flag-transitive $2$-$(v, 5, \lambda)$ designs is on the affine groups and almost simple groups with socle a simple group of Lie type. Hence, we have the following problem, which needs further study.

Problem. Classify $2$-$(v, 5, \lambda)$ designs admitting a flag-transitive, point-primitive automorphism group whose socle is an elementary abelian group or a simple group of Lie type.

In the following, we will first collect some useful preliminary results in Section 2 and then prove Theorem 1.1 in Section 3.

2.   Preliminary results

Before we start the proof of the main result, we first present some necessary preliminary results concerning 2-designs and their automorphisms.

The following facts about the parameters of a 2-design are well known and play a key role in the proof.

Lemma 2.1. [11, Section 2.1] The parameters of a $2$-$(v, k, \lambda)$ design satisfy the following:

(1) $r(k-1) = \lambda(v-1)$;

(2) $bk = vr$;

(3) $b\geq v$ and $r\geq k$;

(4) $r^{2} > \lambda v$.

The following lemma concerns the non-trivial subdegrees of a flag-transitive group $G$ on a 2-design ${\mathcal{D}}$. Note that a subdegree of $G$ is the length of a non-trivial orbit of a point-stabilizer $G_x$, where $x\in{\mathcal{P}}$.

Lemma 2.2. [12, Lemma 1] If $G$ is a group acting flag-transitively on a 2-design, and $d$ is a non-trivial subdegree of $G$, then $r\mid \lambda d$.

In particular, for 2-designs with $k = 5$, by Lemma 2.1 (1) and Lemma 2.2, we have the following immediate corollary.

Lemma 2.3. If $G$ is a group acting flag-transitively on a $2$-$(v, 5, \lambda)$ design, then for each non-trivial subdegree $d$ of $G$, we have $v-1\mid 4d$.

The following lemma is the simplified version of the classification of maximal subgroups of finite alternating groups and symmetric groups. More details shall be found in ^[13], where the authors particularly investigated the maximality of primitive groups, including the groups of affine type, almost simple type, diagonal type, and product action type.

Lemma 2.4. [13, Theorem] Let $G$ be an alternating group ${{{\rm{Alt}}}}(n)$ or a symmetric group ${{{\rm{Sym}}}}(n)$ on a set $\Omega_n$. If $M$ is a maximal subgroup of $G$ with

then one of the following holds:

(1)

acts intransitively on $\Omega_n$, where

both $s$ and $t$ are positive integer, and $s\neq t$;

(2)

acts transitively and imprimitively on $\Omega_n$, where

both $s$ and $t$ are positive integer;

(3) $M$ acts primitively on $\Omega_n$.

Suppose that $G$ is a flag-transitive group of automorphisms of a 2-design ${\mathcal{D}}$. From Lemma 2.1 (4) we easily obtain

by

and

where $(x, B)$ is a flag of ${\mathcal{D}}$. A subgroup $H$ of a group $G$ satisfying

is called large in $G$. Hence flag-transitive groups of a 2-design have large stabilizers. The following lemma for large maximal subgroups of alternating groups and symmetric groups is significant in the proof.

Lemma 2.5. [14, Theorem 2 and Proposition 6.1] Let $G$ be an alternating group ${{{\rm{Alt}}}}(n)$ or a symmetric group ${{{\rm{Sym}}}}(n)$ $(n\geq5)$. If $H$ is a large subgroup and is maximal in $G$ acting primitively on $\Omega_n$, then $(G, H)$ is one of the following:

(1) $(G, H) = ({{{\rm{Sym}}}}(n), {{{\rm{Alt}}}}(n))$;

(2) $G = {{{\rm{Alt}}}}(n)$ and $(G, H)$ is one of the following: $({{{\rm{Alt}}}}(5), D_{10})$, $({{{\rm{Alt}}}}(6), {{{\rm{PSL}}}}_2(5))$, $({{{\rm{Alt}}}}(7), {{{\rm{PSL}}}}_2(7))$, $({{{\rm{Alt}}}}(8), {{{\rm{AGL}}}}_3(2))$, $({{{\rm{Alt}}}}(9), 3^2.{{{\rm{SL}}}}_2(3))$, $({{{\rm{Alt}}}}(9), {{{\rm{P}}}\Gamma{{\rm{L}}}}_2(8))$, $({{{\rm{Alt}}}}(10), {{{\rm{M}}}}_{10})$, $({{{\rm{Alt}}}}(11), {{{\rm{M}}}}_{11})$, $({{{\rm{Alt}}}}(12), {{{\rm{M}}}}_{12})$, $({{{\rm{Alt}}}}(13), {{{\rm{PSL}}}}_3(3))$, $({{{\rm{Alt}}}}(15), A_8)$, $({{{\rm{Alt}}}}(16), {{{\rm{AGL}}}}_4(2))$, $({{{\rm{Alt}}}}(24), {{{\rm{M}}}}_{24})$;

(3) $G = {{{\rm{Sym}}}}(n)$ and $(G, H)$ is one of the following: $({{{\rm{Sym}}}}(5), {{{\rm{AGL}}}}_1(5))$, $({{{\rm{Sym}}}}(6), {{{\rm{PGL}}}}_2(5))$, $({{{\rm{Sym}}}}(7), {{{\rm{AGL}}}}_1(7))$, $({{{\rm{Sym}}}}(8), {{{\rm{PGL}}}}_2(7))$, $({{{\rm{Sym}}}}(9), {{{\rm{AGL}}}}_2(3))$, $({{{\rm{Sym}}}}(10), {{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9))$, $({{{\rm{Sym}}}}(12), {{{\rm{PGL}}}}_2(11))$.

3.   Proof of the main theorem

In this section we prove Theorem 1.1. Throughout the section, we always assume that $G$ is an almost simple group with a socle an alternating group ${{{\rm{Alt}}}}(n)$ with $n\geq5$. Such almost simple groups are known explicitly. It is well known that if $n\ne6$, then $G$ is ${{{\rm{Alt}}}}(n)$ or ${{{\rm{Sym}}}}(n)$. If $n = 6$, then $G = {{{\rm{Alt}}}}(6)$, ${{{\rm{Sym}}}}(6)$, ${\rm M_{10}}$, ${{{\rm{PGL}}}}_2(9)$, or ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$. We first deal with the latter three exceptional cases, ${\rm M_{10}}$, ${{{\rm{PGL}}}}_2(9)$, and ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$, in Section 3.2.

In the second step, we study the cases

and

in Sections 3.3–3.5. Since $G$ acts primitively on $\mathcal{P}$ of the designs, the point-stabilizer $G_x$ for $x\in\mathcal{P}$ is a maximal subgroup of $G$. We note that $G$ has two transitive actions, respectively on $\mathcal{P}$ and $\Omega_n$. According to Lemma 2.4, the groups $G_x$ are divided into three different types with explicit structures according to their actions on $\Omega_n$: intransitive action, imprimitive action, and primitive action. Our strategy of proving the non-existence of or constructing such designs is connecting the action of $G$ on the point set $\mathcal{P}$ of the design and the action of $G$ on the $\Omega_n$. That is the action of $G$ on the coset space of the corresponding maximal subgroups. We shall apply the known degrees and subdegrees in these actions and combine them with the parameter conditions to help us achieve the result. We handle these three types of maximal subgroups in Sections 3.3–3.5, respectively.

3.1.   A procedure of constructing designs

We mention a method (or a procedure) of using the algebraic computer system Magma ^[15] to rule out or construct designs with relatively small parameters $(v, k, \lambda)$. The procedure contains the following steps:

Step 1. Output the group $G$ acting on a set ${\mathcal{P}}$ by its coset action on the corresponding maximal subgroup.

Step 2. Check if $G$ has a subgroup of order $|G_{B}|$. Denote by $J$ such a subgroup as a candidate for the block stabilizer $G_B$.

Step 3. Find the orbits of $J$ with length $k$. Denote such an orbit by $O$ as a candidate for a block $B$.

Step 4. Apply the command ${\tt IsDesign()}$ to check if the incidence structure

forms a 2-design, where

Step 5. For the 2-designs with the same parameters $(v, k, \lambda)$, apply the command ${\tt IsIsomorphic()}$ to check if they are isomorphic to each other.

This procedure will then be applied to rule out designs and construct designs with known parameters $(v, k, \lambda)$ obtained in the proof of the main theorem.

3.2.   The exceptional groups in the case $n=6$

In this section we shall first assume that $G$ has a socle ${{{\rm{Alt}}}}(6)$, but $G$ is neither ${{{\rm{Alt}}}}(6)$ nor ${{{\rm{Sym}}}}(6)$. That is, $G = {\rm M_{10}}$, ${{{\rm{PGL}}}}_2(9)$, or ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$.

Lemma 3.1. Let $G$ be one of ${\rm M_{10}}$, ${{{\rm{PGL}}}}_2(9)$, and ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$, and act flag-transitively on a $2$-$(v, 5, \lambda)$ design ${\mathcal{D}} = (\mathcal{P}, \mathcal{B})$. Let $x\in\mathcal{P}$ and $B\in \mathcal{B}$. Then one of the following holds:

(1) ${\mathcal{D}}$ is a unique $2$-$(10, 5, 8)$ design with

(2) ${\mathcal{D}}$ is a unique $2$-$(10, 5, 16)$ design with

or with

Further, in these cases, $G$ acts 2-transitively on $\mathcal{P}$.

Proof. The maximal subgroups of ${\rm M_{10}}$, ${{{\rm{PGL}}}}_2(9)$ and ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$ have explicit structures, which could be checked in ^[16], for example. We see that ${\rm M_{10}}$ has large maximal subgroups ${{{\rm{AGL}}}}_1(5)$, $Z_3^2$: $Q_8$ and $Z_8$: $Z_2$; ${{{\rm{PGL}}}}_2(9)$ has large maximal subgroups $D_{20}$, $Z_3^2$: $Z_8$ and $D_{16}$; and ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$ has large maximal subgroups $Z_{10}$: $Z_4$, $Z_3^2$: $(Z_8. Z_2)$ and $Z_8.(Z_2\times Z_2)$. We will investigate all these cases by the basic properties of parameters in Lemma 2.1 and the elimination procedure presented in Section 3.1.

First we let

If the point-stabilizer $G_x$ is ${{{\rm{AGL}}}}_1(5)$, then $G$ is 2-transitive on $\mathcal{P}$, with

By the flag-transitivity, we have $r\mid|G_x|$, and so $r\in\{5, 10, 20\}$. None of these satisfies that

(Lemma 2.1 (1)) is an integer. If

then

As

we have $9\mid r$. Note that $r\mid|G_x|$. So $r\in\{9, 18, 36, 72\}$. We indeed obtain 4 admissible parameter sets

For each of these parameters set, the possible order of the block stabilizer is 40, 20, 10, or 5, respectively. Implemented in Magma, we find that there is one subgroup of order 20 (up to conjugacy) such that it has two orbits of length 5, generating two $2$-$(10, 5, 8)$ designs, admitting $G = {\rm M_{10}}$ as their automorphism group. We then check they are isomorphic to each other by Step 5 in Section 3.1.

Similarly, we can apply the same argument for

(as it has the same orders of large maximal subgroups as ${{{\rm{M}}}}_{10}$), and then we see that if

and

then there is exactly one subgroup of order 10 (up to conjugacy) such that it has two orbits of length 5, generating two isomorphic $2$-$(10, 5, 16)$ designs.

At last, for

we can easily rule out the two possible cases in terms of the point-stabilizer isomorphic to $Z_{10}$: $Z_4$ or $Z_8\cdot(Z_2\times Z_2)$, by the basic equation

and

If

then we obtain 4 admissible parameter sets

We find that there is exactly one subgroup of order 10 (up to conjugacy) in $G$, such that it has two orbits of length 5 generating two isomorphic $2$-$(10, 5, 16)$ designs. We also check that this $2$-$(10, 5, 16)$ design is also isomorphic to the design arising from

Further, we notice that

is 2-transitive, see for example [10, Table 9.6.2]. □

In the following, we always suppose that

or ${{{\rm{Sym}}}}(n)$ (satisfying $n\geq5$ by the simplicity of ${{{\rm{Alt}}}}(n)$) in Sections 3.3–3.5. For convenience, we make the following hypothesis:

Hypothesis 1. Suppose that

be a $2$-$(v, 5, \lambda)$ design, and suppose that

or ${{{\rm{Sym}}}}(n)$, acting as a flag-transitive automorphism group of ${\mathcal{D}}$, where $n\geq5$.

3.3.   The action of $G_x$ on $\Omega_n$ is intransitive

In this section, we study the case that the action of $G_x$ on $\Omega_n$ is intransitive. One design is obtained in this case.

Lemma 3.2. Suppose that Hypothesis 1 holds. If $x\in\mathcal{P}$ and $G_x$ is intransitive on $\Omega_n$, then ${\mathcal{D}}$ is a unique $2$-$(21, 5, 12)$ design with

and

or with

and

Further, $G$ acts on $\mathcal{P}$ with rank 3, and the two suborbits have length 10.

Proof. By Lemma 2.4 (1), the structure of $G_x$ is

where

$s$ and $t$ are both positive integers, and $s\neq t$. This is the largest subgroup of $G$ leaving a set of cardinality $s$ (or $t$) invariant. We assume that $s < t$ (equivalently, $s < \frac{n}{2}$), without loss of generality. Since $G$ is flag-transitive, it is easily shown that $G_x$ cannot fix two points in $\mathcal{P}$. In fact, this can also be deduced from the maximality of $G_x$ and that $G$ is not a group of prime order. Further, since $G_x$ is the largest group stabilizing a unique subset of cardinality $s$ in $\Omega_n$, and $G$ acts transitively on all the $s$-subsets of $\Omega_n$ (denoted by $\Omega_n^{\{s\}}$) by its multiple transitivity (at least $(n-2)$-fold), this allows us to identify the point set $\mathcal{P}$ as $\Omega_n^{\{s\}}$. In addition, the action of $G$ acting on ${\mathcal{P}}$ is equivalent to the action of $G$ acting on $\Omega_n^{\{s\}}$. Now, ${\mathcal{D}}$ has

points. Further, one could see, for example, [17, Lemma 3.2] for subdegrees of $G$ in this action: That is, the subdegrees are

where $i = 0, 1, \cdots, s.$ The rank of $G$ in this action is equal to $s+1$. If $s = 1$, then $G$ acts on $\mathcal{P}$ in its natural action, and $\mathcal{D}$ is a complete design, which is not under our consideration. We then separate the subsequent proof into the following two cases.

Case 1. Assume that $s = 2$. Then

Pick two subdegrees

and

According to Lemma 2.3, we get that $\frac{n(n-1)}{2}-1$ divides both $8(n-2)$ and $2(n-2)(n-3)$. It follows that $n = 7$, and so $v = 21$. By Lemma 2.1, one finds

Since $r$ divides $|G|$, $\lambda$ is a divisor of 48. Hence, we then get all the admissible parameters triples $(\lambda, b, r)$. They are

for

and all these are admissible for

except

Now, by the elimination procedure presented in Section 3.1, we obtain a unique $2$-$(21, 5, 12)$ design with flag-transitive groups ${{{\rm{Alt}}}}(7)$ and ${{{\rm{Sym}}}}(7)$, and

respectively. Further, $G$ has rank 3 on $\mathcal{P}$, with the two suborbits of length 10.

Case 2. Assume that $s\geq3$. Then

Pick

By Lemma 2.3, we obtain $v-1\mid 4s(n-s)$. We notice a fact that

Since $s\geq3$, we have

Note that

for $n\geq9$. Moreover, it is easily known that

We have

for $n\geq9$, which contradicts $v-1\mid 4s(n-s)$. Hence, $n$ is either 7 or 8. If $n = 7$, then $s = 3$ and $t = 4$, and so

and

which contradicts

If $n = 8$, then $s = 3$ and $t = 5$, and so

and

which is again impossible. □

3.4.   The action of $G_x$ on $\Omega_n$ is transitive and imprimitive

In the following, according to Lemma 2.4 (2), we tackle the case that the point stabilizer $G_x$ is imprimitive on $\Omega_n$. We find that there are two admissible designs and determine the corresponding automorphism groups.

Lemma 3.3. Suppose that Hypothesis 1 holds. If $x\in\mathcal{P}$ and the point-stabilizer $G_x$ is imprimitive on $\Omega_n$, then one of the following holds:

(1) ${\mathcal{D}}$ is a unique $2$-$(10, 5, 8)$ design with $G = {{{\rm{Alt}}}}(6)$, $G_x\cong ({{{\rm{Sym}}}}(3)\wr{{{\rm{Sym}}}}(2))\cap {{{\rm{Alt}}}}(6)$, and $G_B\cong D_{10}$;

(2) ${\mathcal{D}}$ is a unique $2$-$(10, 5, 16)$ design with $G = {{{\rm{Sym}}}}(6)$, $G_x\cong {{{\rm{Sym}}}}(3)\wr{{{\rm{Sym}}}}(2)$, and $G_B\cong D_{10}$.

Further, in these two cases, $G$ acts 2-transitively on $\mathcal{P}$.

Proof. Suppose that

acting as the largest imprimitive group on $\Omega_n$ (in the natural action of $G$). Then $\Omega_n$ has a point-partition

with $|\Gamma_i| = s$ and $n = st$, which is invariant under the action of $G_x$. It is obvious that

According to the analysis in [18, Section 3], the group $G_x$ has only one invariant non-trivial point-partition of $\Omega_n$. Denote by $P_{\Omega_n}^{s}$ the set of all the partitions of $\Omega_n$ into $t$ parts, each of length $s$. Then it will be straightforward to verify that the action of $G$ on ${\mathcal{P}}$ is equivalent to the action of $G$ on $P_{\Omega_n}^{s}$ (in the natural induced action of $G$ on $\Omega_n$). Hence, we can identify $\mathcal{P}$ with $P_{\Omega_n}^{s}$ and identify the action of $G$ on ${\mathcal{P}}$ with the action of $G$ on $P_{\Omega_n}^{s}$. Thus,

which could be calculated directly, or equals $\frac{|G|}{|G_x|}$ as $G$ acts transitively on $P_{\Omega_n}^{s}$. Then, we have

If $s = 2$, see, for example, [19, Section 3.2], and we know that there exist suborbits of $G$ (acting on the coset space of $G_x$ in $G$) with degree

where $j = 2, \cdots, t$. For $s\geq3$, there exist subdegrees

where $j = 2, \cdots, t$.

Case 1. Assume that $s = 2$. Then it would be straightforward to see that

Since

we have $t\geq3$. Pick subdegree

By Lemma 2.3, we have

If $t\geq4$, then

which is a contradiction. If $t = 3$, then

does not divide

a contradiction.

Case 2. Assume that $s\geq3$. Pick subdegree

By Lemma 2.3, we have $v-1\mid 2s^2t(t-1)$. It is not hard to see that

This implies

So

If $t\geq3$, then

Only

is possible. Now

and

which contradicts $v-1\mid2s^2t(t-1)$. Thus, $t = 2$. Now

We obtain

We tackle these cases one by one:

Subcase 1. If $s = 3$, then

and

This case satisfies $v-1\mid4s^2$. Implemented by Magma, we indeed find a unique $2$-$(10, 5, 8)$ design admitting a flag-transitive group ${{{\rm{Alt}}}}(6)$ and a unique $2$-$(10, 5, 16)$ design admitting a flag-transitive group ${{{\rm{Sym}}}}(6)$. Both of the block stabilizers are

We also notice that

is 2-transitive (see [10, Table 9.62]).

Subcase 2. If $s = 4$, then

and

This does not satisfy $v-1\mid4s^2$.

Subcase 3. If $s = 5$, then $v = 126$, and $4s^2 = 100$. We have

a contradiction. If $s = 6$, then $v = 462$, and $4s^2 = 144$. We have

a contradiction. If $s = 7$, then $v = 1716$, and $4s^2 = 196$. If $s = 8$, then $v = 6435$, and $4s^2 = 256$. If $s = 9$, then $v = 24310$, and $4s^2 = 324$. For all cases with $s\geq5$, we have

a contradiction to $v-1\mid4s^2$. □

3.5.   The action of $G_x$ on $\Omega_n$ is primitive

In this final section, we deal with the case of Lemma 2.4 (3), that is, $G_x$ is a primitive group on $\Omega_n$ in the natural action of $G$. We obtain three different designs, as listed in the lemma below.

Lemma 3.4. Suppose that Hypothesis 1 holds. If $x\in\mathcal{P}$ and the point-stabilizer $G_x$ of $G$ acts primitively on $\Omega_n$, then we obtain the following:

(1) ${\mathcal{D}}$ is a unique $2$-$(15, 5, 4)$ design with $G = {{{\rm{Alt}}}}(7)$, $G_x\cong {{{\rm{PSL}}}}_3(2)\cong{{{\rm{PSL}}}}_2(7)$, and $G_B\cong{{{\rm{Alt}}}}(5)$;

(2) ${\mathcal{D}}$ is a unique $2$-$(15, 5, 12)$ design with $G = {{{\rm{Alt}}}}(7)$, $G_x\cong {{{\rm{PSL}}}}_3(2)\cong{{{\rm{PSL}}}}_2(7)$, and $G_B\cong Z_5$: $Z_4$;

(3) ${\mathcal{D}}$ is a unique $2$-$(15, 5, 16)$ design with $G = {{{\rm{Alt}}}}(8)$, $G_x\cong {{{\rm{AGL}}}}_3(2)$, and $G_B\cong{{{\rm{Sym}}}}(5)$.

Further, in these three cases, $G$ acts 2-transitively on $\mathcal{P}$.

Proof. Since a flag-transitive group of automorphisms has large point-stabilizer, the group $G$ and the possible point-stabilizer $G_x$ are listed in Lemma 2.5. In the meantime, the possible number of points

equals $\frac{|G|}{|G_x|}$. By the five steps of the procedure of elimination and construction given in Section 3.1, and with the implementation in Magma, we see that only the two cases

and (Alt(8), AGL(3, 2)) yield flag-transitive 2-designs with $k = 5$. For

we obtain two designs with parameters

and the block stabilizers are

and

respectively. For

we obtain a unique design with parameters

with

Further, we note that

and ${{{\rm{Alt}}}}(8)$ are 2-transitive on $\mathcal{P}$ (see [10, Table 9.62]). □

Proof of Theorem 1.1. In Lemmas 3.1–3.4, we inspect the three different types of maximal subgroups of $G$, as listed in Lemma 2.4. Further, by Step 5 in Section 3.1, we check that the $2$-$(10, 5, 8)$ designs in Lemmas 3.1 and 3.3 are isomorphic. Besides, the $2$-$(10, 5, 16)$ designs in Lemmas 3.1 and 3.3 are also isomorphic. Therefore, there are a total of six different 2-designs with block size 5, admitting flag-transitive groups whose socle is a simple alternating group. □

4.   Conclusions

In Section 3, we prove the main result of this paper: give a complete classification of 2-designs with block size 5, admitting a flag-transitive automorphism group ${{{\rm{Alt}}}}(n)$, ${{{\rm{Sym}}}}(n)$, ${{{\rm{M}}}}_{10}$, ${{{\rm{PGL}}}}_2(9)$, or ${{{\rm{P}}}\Gamma{{\rm{L}}}}_2(9)$. We think that the techniques and methods used in the proof of the current paper might also be useful in the study of 2-designs satisfying other conditions, even in other objects such as transitive graphs.

Author contributions

Jiaxin Shen: conceptualization, writing—original draft preparation, writing—review and editing; Yuqing Xia: writing—original draft preparation, writing—review and editing. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank anonymous reviewers for providing valuable comments and suggestions that help improve the paper.

This research was funded by the National Natural Science Foundation of China with grant numbers 12201469 and 12471015, and by the Special Foundation in Key Fields for Universities of Guangdong Province grant number 2022ZDZX1034.

Conflict of interest

All authors declare no conflicts of interest in this paper.