#### Abstract

We describe a construction of primitive 2-designs and strongly regular graphs from the simple groups , and . The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of , and . In addition, from the group , we construct 2-designs with parameters and having the full automorphism group isomorphic to .

#### 1. Introduction

Questions about combinatorial structures related to finite groups arose naturaly in studying of the groups. Studies on the interplay between finite groups and combinatorial structures have provided many useful and interesting results. From a geometric point of view, the most interesting designs are generally those admitting large automorphism groups. Famous Witt designs constructed from Mathieu groups have been discovered in 1930βs (see [1, 2]), and for some further construction of combinatorial structures from finite groups, we refere the reader to [3β5]. A construction of 1-designs and regular graphs from primitive groups is described in [6] and corrected in [7]. The construction employed in this paper is described in [8], as a generalization of the one from [6, 7].

An incidence structure is an ordered triple where and are nonempty disjoint sets and . The elements of the set are called points, the elements of the set are called blocks, and is called an incidence relation. If , then the incidence structure is called symmetric. The incidence matrix of an incidence structure is a matrix where and are the number of blocks and points, respectively, such that if the point and block are incident, and otherwise. An isomorphism from one incidence structure to another is a bijective mapping of points to points and blocks to blocks which preserves incidence. An isomorphism from an incidence structure onto itself is called an automorphism of . The set of all automorphisms forms a group called the full automorphism group of and is denoted by .

A design is a finite incidence structure satisfying the following requirements: (1),(2)every element of is incident with exactly elements of ,(3)every elements of are incident with exactly elements of .

A design is called a block design. A design is called quasisymmetric if the number of points in the intersection of any two blocks takes only two values. If , then the design is called symmetric. A symmetric design is called a projective plane.

An incidence structure has repeated blocks if there are two blocks incident with exactly the same points. An incidence structure that has no repeated blocks is called simple. All designs described in this paper are simple.

Let be a finite incidence structure. is a graph if each element of is incident with exactly two elements of . The elements of are called vertices, and the elements of are called edges. Two vertices and are called adjacent or neighbors if they are incident with the same edge. The number of neighbors of a vertex is called the degree of . If all the vertices of the graph have the same degree , then is called *k*-regular. Define a square -matrix labelled with the vertices of in such a way that if and only if the vertices and are adjacent. The matrix is called the adjacency matrix of the graph .

A graph is called a strongly regular graph with parameters and denoted by SRG if is -regular graph with vertices and if any two adjacent vertices have common neighbors and any two nonadjacent vertices have common neighbors.

Let *x* and *y * be the two cardinalities of block intersections in a quasi-symmetric design . The block graph of has as vertices the blocks of and two vertices are adjacent if and only if they intersect in points. The block graph of a quasi-symmetric design is strongly regular. In a design which is not a projective plane, two blocks intersect in 0 or 1 points; therefore, the block graph of this design is strongly regular (see [9]).

Let be a symmetric design which possesses a symmetric incidence matrix with 1 everywhere on the diagonal. Then, the matrix is an adjacency matrix of a strongly regular graph with parameters (see [9]) and .

Let be a simple group and be a maximal subgroup of . The conjugacy class of is denoted by . Obviously, , so . Denote the elements of the conjugacy class by , .

In this paper, we consider block designs constructed from the linear group , strongly regular graphs constructed from the unitary group , and block designs and strongly regular graphs constructed from the symplectic group .

is the simple group of order 372000, and it has five distinct classes of maximal subgroups: , , , and . We define incidence structures on the elements of conjugacy classes of the maximal subgroups of ; that is, points and blocks are labelled by elements of conjugacy classes of the maximal subgroups of .

is the simple group of order 13685760, and it has six distinct classes of maximal subgroups: , , , , , and . We define strongly regular graphs whose vertices are labelled by elements of conjugacy classes of the maximal subgroups of .

is the simple group of order 1451520, and it has eight distinct classes of maximal subgroups: , , , , , , , and . We define incidence structures on the elements of conjugacy classes of the maximal subgroups of and strongly regular graphs whose vertices are labelled by elements of conjugacy classes of the maximal subgroups of .

Generators of groups , , and and their maximal subgroups are available on the Internet: http://brauer.maths.qmul.ac.uk/Atlas/.

In this paper, we describe a construction of primitive block designs with parameters , , , , , , , , , , , , , , and and strongly regular graphs with parameters , , , , , , and . The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of the simple groups , and . In addition, from the group , we construct 2-designs with parameters and having the full automorphism group isomorphic to .

The graphs described in this paper have been previously known, since they can be constructed as rank 3 graphs. For more details on rank 3 graphs, we refer the reader to [9]. Some of the constructed block designs on 31 points have large number of blocks, and we did not find a record that they have been previously studied, although these designs can be constructed from . The constructed designs with parameters , , , , and are isomorphic to the already known designs. However, we did not find an evidence that the constructed designs with parameters , , and are isomorphic to the already known objects.

The paper is organized as follows: in Section 2, we describe the method of construction of primitive designs and graphs used in this paper, Section 3 describes block designs on 31 points constructed from the group , Section 4 gives strongly regular graphs constructed from the group , and Section 5 describes the group as block designs and strongly regular graphs. At the end of the paper, we give a list of the constructed designs and strongly regular graphs and their full automorphism groups.

For basic definitions and group theoretical notation, we refer the reader to [10, 11].

#### 2. The Construction

The following construction of symmetric 1-designs and regular graphs is presented in [6, 12].

Theorem 2.1. *Let G be a finite primitive permutation group acting on the set of size . Let , and let be an orbit of the stabilizer of . If and, given , , then forms a symmetric design. Further, if is a self-paired orbit of , then is a regular connected graph of valency , is self-dual, and G acts as an automorphism group on each of these structures, primitive on vertices of the graph and on points and blocks of the design.*

In [8], we introduced a generalization of the above construction. This generalization, presented below in Theorem 2.2, allows us to construct 1-designs that are not necessarily symmetric and stabilizers of a point and a block are not necessarily conjugate.

Theorem 2.2. *Let be a finite permutation group acting primitively on the sets and of size and , respectively. Let , , and let be the -orbit of and be the -orbit of . If and
**
then is a design with m blocks and G acts as an automorphism group, primitive on points and blocks of the design.*

The construction of a design described in Theorem 2.2 can be interpreted in the following way: (i)the point set is ,(ii)the block set is ,(iii)the block is incident with the set of points .

Let a point be incident with a block . Then, ; hence, there exists such that . Therefore,

If a point is incident with the block , then . If the set contains ) -conjugacy classes, where is the number of -orbits on , then each conjugacy class corresponds to one -orbit and the incidence relation in the design can be defined as follows:

(i) the block is incident with the point if and only if is conjugate to .

Similarly, if the set contains isomorphism classes, then the incidence in the design can be defined as follows:

(ii) the block is incident with the point if and only if .

In the construction of the design described in Theorem 2.2, instead of taking a single -orbit, we can take to be any union of -orbits.

Corollary 2.3. *Let be a finite permutation group acting primitively on the sets and of size and , respectively. Let and , where are representatives of distinct -orbits. If and
**
then is a design with m blocks and G acts as an automorphism group, primitive on points and blocks of the design.*

*Proof. *Clearly, the number of points is , since the point set is . Further, each element of consists of elements of .

The set is a union of -orbits, so , where is the setwise stabilizer of . Since is primitive on , is a maximal subgroup of , and therefore . The number of blocks is

Since *G* acts transitively on and , the constructed structure is a 1-design, hence , where each point is incident with blocks. Therefore,
and consequently
It follows that

*Remark 2.4. *In the construction of graphs described in Theorem 2.1, we can define the set of edges as a union .

The construction described in Corollary 2.3 gives us all designs that admit primitive action of the group on points and blocks.

Corollary 2.5. *If the group acts primitively on the points and the blocks of a 1-design , then can be obtained as described in Corollary 2.3, that is, such that is a union of -orbits.*

*Proof. *Let be any block of the design . *G* acts transitively on the block set of the design , hence . Since acts primitively on , the stabilizer is a maximal subgroup of . fixes , so is a union of -orbits.

We can obtain a 1-design by defining the incidence in such a way that the point is incident with the block if and only if where .

Let *G* be a simple group and and be maximal subgroups of *G*. The stabilizer of any element , , is the maximal subgroup , hence *G* acts primitively on the class , , by conjugation and

Corollary 2.3 allows us to define a 1-design on conjugacy classes of the maximal subgroups and of a simple group . Let us denote the elements of by and the elements of by .

We can construct a 1-design such that (i)the point set of the design is , (ii)the block set is , (iii)the block is incident with the point if and only if , , where .

Let us denote a 1-design constructed in this way by .

Similarly, from the conjugacy class of a maximal subgroup *H* of a simple group *G*, one can construct regular graph in the following way: (i)the vertex set of the graph is , (ii)the vertex is adjacent to the vertex if and only if , , where .

We denote a regular graph constructed in this way by .

*Remark 2.6. *Let be an automorphism of a finite group . Then, the design is isomorphic to , and the graph is isomorphic to .

#### 3. Block Designs on 31 Points Constructed from the Group

Let *G* be a group isomorphic to the linear group .

Using GAP (see [13]), one can check that or for all . Further, for every ,

Let us define sets , . For every , , , the set has exactly one element. That proves that the incidence structure is the unique symmetric design (see [14]), that is, the projective plane . The full automorphism group of the design has 372000 elements, and it is isomorphic to the group .

Let be a maximal subgroup in . The conjugacy class has 3100 elements. One can verify using GAP that is isomorphic to , , or , for all . Using GAP (see [13]), we obtain the following:(1), for all , ,(2), for all , , ,(3), for all , ,(4), for all , , ,(5), for all , ,(6), for all , , .

It follows from (1) and (2) that the structure is a block design . The blocks of are conics in .

(3) and (4) imply that the structure is a block design . The blocks of are interior points of conics in .

Further, (5) and (6) show that the structure is a block design . The blocks of are exterior points of conics in .

The full automorphism group of designs , , and has 372000 elements and is isomorphic to the group .

Let be a maximal subgroup in . The conjugacy class has 3875 elements. Using GAP (see [13]), one can check that is isomorphic to , , or , for all . In the similar way as above, we obtain the following results:(i) is a design. is isomorphic to the design constructed from the symmetric design in such a way that the blocks are all triples of noncollinear points. (ii)The incidence structure is a block design . The blocks of are unions of sides of triangles in . (iii) is a block design . The blocks of are unions of sides of triangles in , including the corners.

The full automorphism group of , , and is isomorphic to .

Six designs isomorphic to , , , , , and can be obtain in the same way as described in this paper using the maximal subgroup instead of . This is a consequence of the Remark 2.6 and the fact that there exists an automorphism of which fixes , and , setwise and acts as a transposition which maps onto .

We thank an anonymous referee of an earlier version of this paper for suggesting the construction of the designs , , , , and from triangles and conics in .

#### 4. Strongly Regular Graphs Constructed from the Group

In this section, we describe structures constructed from a simple group isomorphic to the unitary group .

The intersection of two different elements , , denoted by , is isomorphic to (i) or , (ii) or , (iii) or , (iv), , , , or .

Applying the method described in Section 2, we obtain the following results: (i)the graph is a strongly regular graph with parameters , (ii)the graph is a strongly regular graph with parameters , (iii)the graph is a strongly regular graph with parameters , (iv)the graph is a strongly regular graph with parameters .

The full automorphism groups of the graphs , , and are isomorphic to the group . is a rank 3 group on 165, 176, and 297 points, and , , and are rank 3 graphs of the group .

The full automorphism group of the graph is of order 18393661440 and is isomorphic to the group . Although acts as a rank 7 group on 1408 points, the graph can be constructed as a rank 3 graph from the Fischer group .

#### 5. Structures Constructed from the Group

In this section, we consider the structures constructed from a simple group isomorphic to the symplectic group .

The intersection of two different elements and , denoted by , is isomorphic to (we introduce only the intersection of elements of conjugacy classes that give rise to a strongly regular graph or a block design) (i) or , (ii) or , (iii) or , (iv) or , (v) or , (vi) or , (vii) or , (viii) or , (ix) or , (x), , or .

Applying the method described in Section 2, we obtain the following results: (i)the incidence structure is a block design with parameters , (ii)the incidence structure is a block design with parameters , (iii)the incidence structure is a block design with parameters , (iv)the incidence structure is a block design with parameters , (v)the incidence structure is a block design with parameters , (vi)the incidence structure is a block design with parameters , (vii)the incidence structure is a block design with parameters , (viii)the incidence structure is a block design with parameters , (ix)the graph is a strongly regular graph with parameters , (x)the graph is a strongly regular graph with parameters , (xi)the graph is a strongly regular graph with parameters .

The full automorphism groups of designs , , and and the full automorphism group of the graph are isomorphic to .

Graphs and have as the full automorphism group. acts as a rank 3 group on 120 and 135 points, and graphs and can be constructed as rank 3 graphs from the orthogonal group . On the other hand, the group acts as a rank 3 group on 120 points and as a rank 4 group on 135 points. Therefore, the graph can also be constructed as a rank 3 graph from the .

The design is a quasi-symmetric SDP design isomorphic to the design described in [15, 16]. Its block graph is isomorphic to the complement of the graph .

Furthermore, from the design on 28 points, one can construct two designs on 28 points, one isomorphic to the design and one isomorphic to the design . The design on 28 points, whose blocks are unions of three disjoint blocks of the design such that stabilizer of that union under the action of the automorphism group is isomorphic to the maximal subgroup , is isomorphic to the design . On the other hand, unions of three blocks intersecting in exactly one point of the design such that stabilizer of that union under the action of the group is isomorphic to the maximal subgroup are blocks of a design isomorphic to the design .

The design is also a quasi-symmetric design, isomorphic to the design described in [15, 16]. Its block graph is isomorphic to the graph .

The ovals (sets of 6 points, no three collinear) of the design form a block design with parameters isomorphic to the design . Furthermore, the group has a subgroup isomorphic to the group . The group acts on the set of all ovals of the design in four orbits of size 84 and one of these orbits forms the block set of a block design with parameters having as the full automorphism group. The design is isomorphic to the design described in [17].

Unions of four disjoint blocks of the design such that stabilizer of that union under the action of the automorphism group is isomorphic to the maximal subgroup are blocks of a design isomorphic to the complement of the design .

The design is the point-hyperplane design in the projective geometry , and its full automorphism group is isomorphic to . possesses a symmetric incidence matrix with 1 everywhere on the diagonal, and therefore it gives rise to a strongly regular graph with parameters () which is isomorphic to the graph .

The graph can also be constructed from the design . Any two blocks of intersect in 1, 2, or 4 points. The graph which has as its vertices the blocks of , two vertices being adjacent if and only if the corresponding blocks intersect in one point, is isomorphic to .

Let be a subgroup of the group and let and be conjugacy classes of maximal subgroups and under the action of the group . The intersection of , , and , , is isomorphic either to or . Let be an incidence structure whose points are labeled by the elements of the class and whose blocks are labeled by the elements of the class , point and block being incident if and only if the intersection of corresponding elements of conjugacy classes is isomorphic to the group . The structure is isomorphic to the block design with parameters having as the full automorphism group. Further, let , be an element of the class . The incidence structure whose points are labeled by the elements of the class and whose blocks are labeled by the elements of the class , point and block being incident if and only if the intersection of corresponding elements of conjugacy classes is isomorphic to the group , is isomorphic to the Hermitian unital with parameters having as the full automorphism group. We conclude that the designs and are subdesigns of the design . One can construct from the group the design isomorphic to the design (see [15]).

The HΓΆlz design *H(q)* of order *q* is a design with parameters which is a union of the Hermitian unital with parameters and a design whose blocks are arcs in the unital. To prove that design is isomorphic to the design , we constructed design , by the method described in [18], as the support design of the dual code of the code spanned by the incidence vectors of the design . Constructed design is isomorphic to the design .

From(i), (ii), (iii), (iv),

one can conclude that, in order to construct designs , , , and , corresponding maximal subgroups need not to be conjugate by the elements of the whole group . The conjugation by the elements of the subgroup *U* is sufficient to obtain the desired structures.

In Table 4 we give a list of the constructed designs and strongly regular graphs and their full automorphism groups. Table 1, Table 2, and Table 3 give a list of maximal subgroups of the groups , , and , respectively.