Abstract
The focus of this study is to classify flag-transitive 2-designs. We have come to the conclusion that if is a nontrivial 2-design having block size 5 and is a two-dimensional projective special linear group which acts flag-transitively on with (mod 4), then is a 2-(11, 5, 2) design, a 2-(11, 5, 12) design, a 2- design with (mod 4) or a 2- design with (where is an even).
1. Introduction
In the year of 1987, Davies [1] drew a conclusion that if a 2- design has a point-imprimitive flag-transitive automorphism group , then given the value of , there is a superior limit on the block size . We can easily arrive at a conclusion that the existence of the pairs is not infinite only if is fixed, where acting on is point-imprimitive and flag-transitive. Meanwhile, the hidden meaning is that, as long as we set fixed, there are only a finite number of such designs. Still further, on the basis of the proof of ([2], Proposition 4.1), one proving that for a nontrivial point-imprimitive and flag-transitive 2- design , its block size . Without a doubt, we have that the flag-transitive automorphism group for our paper of the 2- design is definitely point-primitive. The O’Nan–Scott theorem shows that one of the following is doomed to hold for any finite primitive permutation group (for more details, see [3]):(a) is of almost simple type(b) is of affine type(c) is of simple diagonal type(d) is of product type(e) is of twisted wreath product type
In order to solve the problem more completely, we narrow down the scope of our problem; here, we only consider the case (a) and is a two dimensional projective special linear group .
For (mod 4), the group acts 3-homogeneously on the one-dimensional projective line. We immediately inferred that if is a set consisting of some -subsets of the projective line , then is a 3- design for some in the condition that is a union of -orbits with . Thereafter that is why becomes a coveted treasure when it comes to construct a 3-design. For instance, through the way acts on the 4-element subsets of the one-dimensional projective line, the authors of [4] determined all 3-designs of with block size 4. By using the same processing method, Keranen and Kreher [5] completely solved such designs of block size 5. On the other hand, for (mod 4), Keranen et al. [6] has completely determined all quadruple systems admitting as their automorphism group. Unfortunately, the 3- designs with the automorphism group remain uncertain.
Many scholars have turned to another direction of exploration, focusing on 2-designs whose automorphism group is the projective group , which acts transitively on the flags. Here, flags are point-block pairs such that . In 1986, Delandtsheer [7] discovered a 2- design with (it is usually called Witt-Bose-Shrikhandle space ) on the way to classify flag-transitive 2- design. Three savants of literature [8] are absorbed in symmetric designs and give them a complete classification. Zhan and Zhou [9], in 2018, dealt with the situation where is a nonsymmetric design with parameters and that are coprime. The most classification of 2- designs, characterized by as a flag-transitive automorphism group, is presented in [10], offering a complete and elegant classification of such designs.
Our paper aims to studying 2-designs with block sizes of 5 and flag-transitive automorphism groups that are the two-dimensional special linear projective group , where (mod 4). Obviously, it is of great significance to contribute to fully classify the 2- designs permitting a flag-transitive automorphism group. Furthermore, the main outcomes of our paper are displayed as follows.
Theorem 1. Assume that is a nontrivial 2-design with block size 5. Let act flag-transitively on with (mod 4). Then one of the following conclusions is proved to be tenable:(1) is a 2-(11,5,2) design or a 2-(11,5,12) design with (2) is a 2- design with or 59 (mod 60)(3) is a 2- design with and is an even integer
In order to have a better understanding for the proving process of Theorem 1 in the third section, we will demonstrate some basic concepts and general principles which will be employed during the process of proving our conclusion.
2. Notation and Preliminaries
Assuming that is a set containing blocks and is a set containing points, the pair points to a 2-design that satisfies every block including points, 2 different points are exactly comprised in blocks, and a given point is relative to blocks. In general, we handle the case , where is called nontrivial. Particularly, a design is said to be symmetric if the total number of blocks in is equal to the total number of points; otherwise, it is called nonsymmteric. A 2- design is often called a finite linear space when . For the study of a 2-design, the following lemma is almost involved for each time.
Lemma 2 (see ([4], 1.2, 1.9)). The following properties hold for a 2-design:(i)(ii)
More often than not, we use to represent the full automorphism group of a design . That is, is the group in particular to those composed of all automorphisms of , where an automorphism of refers to a permutation that can permutate not only the point set but also the block set . That is to say, when is an automorphism group of , any element of must belong to , in short, . If has a primitive action on point set and a transitive action on flag set, then we say design is point-primitive and flag-transitive, respectively.
Take any block , and thereafter is the setwise stabilizer. Below we will introduce a classical and commonly used verdict about them.
Lemma 3. For a 2-design , let , and be the flag set of . If , then the two statements given below are interchangeable:(i) acts flag-transitively on (ii) is a transitive group of , and has a transitive action on
The following lemma adds the finishing touch to the proof of our paper. Literature [1] demonstrates the proof in detail and helps us to understand it better.
Lemma 4. Provided that acts on a 2-design flag-transitively, let be a subdegree of with ; then divides .
From ([4], Theorem 6) and ([5], Theorem 3.3), we can reach the following result which is of most importance through a simple calculation.
Lemma 5. Let with (mod 4), and be a 5-element subset of one-dimensional projective line. Then the following two situations are to be true:(1)If (mod 4), then is one of with , with (mod 60) or with (mod 60)(2)If , then is one of with , with or with even
The following lemma displays the correlation between -transitive and 2-transitive which lies in ([11], Theorem 1.2). It is a special case that will arise in our paper discussion.
Lemma 6. Let be a finite almost simple group, and acts on a set with size -transitively. Then the following conclusions are true:(i) has a 2-transitive action on (ii) where , and each nontrivial subdegree of is
3. Proof of Theorem 1
From the information provided above, we have known that if is a 2-design with and is a flag-transitive group, then must be a point-primitive group. If is the set keeping a certain point in stationary, then must be one of the maximal subgroups in . In the following convention, rank is denoted by the rank of .
Proposition 7. Suppose that is flag-transitive for which is a 2-design with block size 5. Then rank .
Proof. Following Lemma 2, . Assuming is a subdegree of , then combining with Lemma 4, we can easily derive that . Consequently, . As a result, the possible value of the rank of is 2, 3, 4, or 5. Moreover,(i)If rank , then has subdegrees or (ii)If rank , then has subdegrees (iii)If rank , then has subdegrees As mentioned above, the two-dimensional projective special linear group acts doubly transitively on the one-dimensional projective line. It is for this reason that if is a -orbit on 5-subsets of the one-dimensional projective line, then is destined to be the set of block of a 2- design for some . At the same time, it can be easily seen that is deemed to be the block-transitive automorphism group of such designs.
Proposition 8. Let be a nontrivial 2- design. Let be a flag-transitive automorphism group of with (mod 4). Then rank , and one of the following three statements is to be true:(1) has parameters or (11,12), and (2) has parameters with or 59 (mod 60)(3) has parameters with , where is an even
Proof. Suppose that rank ; then is a -transitive group by Proposition 7. According to Lemma 2(ii), we get . Thus, , a contradiction.
Assume that rank . If is maximal subgroup of , then the subdegrees of the representation on the cosets of can be found in [9, 12]. Obviously, we know that has no subdegrees .
Suppose that rank . For the case has subdegrees , is a -transitive group. Similarly, Lemma 2(ii) yields . Thus, , a contradiction. Again by [9, 12], we reach that has no subdegrees .
Now we discuss the only remaining circumstance that rank ; then or . We will now examine these three cases. Case (1): with degree 7. Here the order of is 168 and note that its representations are Clearly, is a doubly transitive group of the point set = {1, 2, …, 7}. Let ; then is a block-transitive 2-design such that its block number as is 2-transitive. Easy calculation shows that the setwise stabilizer has three orbits on as follows: Thus, is impossible to admit as its flag-transitive automorphism group by Lemma 3. Case (2): with degree 11. There are two inequivalent 2-transitive permutation representations of which possess degree 11, namely, and . Here we will confine our discussion to the case , and another case can be discussed similarly. Suppose that is a Sylow 5-subgroup of ; then partitions the 10 points into two orbits with size 5 and fixes one point, and we denote them by and , respectively. Now, there is exactly one conjugacy class of subgroups isomorphic to alternating group partitioning the 11 points into two orbits of lengths 5 and 6. Let be the representative of the conjugacy class such that . So partitions the 11 points into two orbits of length 5 and 6, respectively. It is safe to assume, without losing generality, that is the -orbit of length 5. Then and is the flag-transitive 2- symmetric design by Lemma 3 and the doubly transitivity of . Note that is a Hadamard design of order 3. Finally, is properly contained in the doubly transitive orbit of length 6; then , and hence is the flag-transitive 2- design again by the doubly transitivity of and Lemma 3. Case (3): with degree . If (mod 4), from Lemma 5(1), there exist 5-element subsets , , and such that , and . Lemma 3 yields that can be divided by 5, where is a block. Thus, we conclude that or cannot be the block set of as and . Let ; then is a 2-design withTherefore, Lemma 2 yields that has parameters . Moreover, acts point-transitively on . Then by what we have already shown, is a transitive permutation group on the flags.
Now, let . Because of the nontriviality of , we should restrict . Similarly, from Lemma 5(2), there exist 5-element subsets , , and such that , , and . By Lemma 3, we infer that or cannot be a base block of a flag-transitive design for and . Set , and we conclude that is a 2-design withNow the result provided in Lemma 2 shows again that the 2-design has parameters . Since , we have that acts point-transitively on by ([13], Lemma 16). Accordingly, acting on is flag-transitive [14, 15].
Finally, Theorem 1 can be obtained from Proposition 8.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by the National Natural Science Foundation of China (no. 12361004), Natural Science Foundation of Jiangxi Province (no. 20224BAB211005), and Technology Project of Jiangxi Education Department (GJJ2200669) and Natural Science Foundation of Henan Province of China (no. 242300421688).