A Rank Five Geometry on the Mathieu Group

 

Joan Schunck

 

MATH 6221

 

Fall 2001

The Mathieu Group is a 5-transitive group of permutations which may be constructed in the following way.

 

Let      

           

            .

Then     .

 

 is noteworthy because

·        It is the only known 5-transitive group except for the symmetric and alternating groups of degree 5.

·        It can be viewed as a hexad system  acts on the given 12 symbols so that they may be combined into 132 hexads (or sextuplets). Any given set of 5 of the symbols will appear in exactly one hexad.  In other words, if we view the symbols as the points of a geometry, and the hexads as the blocks of a geometry, each block will be uniquely determined by a set of 5 points. [Carmichael 431]

 

We now construct a rank five geometry on  .

 

We call a geometry a t-(v, k, λ) design (with t, v, k, and λ positive integers, 2 ≤ tk < v), if the number of points in the geometry is v, every block is incident to k points, there are t distinct points incident to λ blocks, and blocks are determined by their point sets.  If λ = 1, then the geometry is called a Steiner system and is denoted S(t, k, v). [Pasini 23]

 

Consider S(5, 8, 24), the Steiner system for , another Mathieu group, which has 24 points, 8 points per block, and 5 points incident to a unique block.  Let Ω represent the point set of this geometry and let be a partition of Ω into two disjoint sets of 12 points each, called dodecads.  The stabilizer G of A is isomorphic to .

Let  denote an object set with an incidence relation.  The types of objects in  are as follows.

·        The objects of type 0 are the points of A. Clearly, there are 12 such points.

·        The objects of type 4 are the points of B.  Again, there are 12 such points.

·        On the 12 points of A, take the Steiner system S(5, 6, 12).  The stabilizer of a hexad in A stabilizes a duad in B.  An object of type 3 is the union of a hexad in A and its corresponding duad in B.  As discussed above,  produces 132 hexads, so that there are 132 objects of this type.

·        On the 12 points of B, take the Steiner system S(5, 6, 12), and define objects of type 1 as the analogous union of hexads in B and corresponding duads in A.  Again, there are 132 objects of this type.

·        Now take a triad  in A and its stabilizer in .  Consider the 12 points of B to be lines in the affine plane AG(2, 3).  There are four partitions of the set of points by three lines in AG(2, 3).  To each partition corresponds a triad  in B.  An object of type 2 is the union of such a triad  in A and a triad  in B.  There are  objects of this type.

 

 

To each type of object in  we now associate its stabilizer in .

·        Objects of type 0 and 4 are stabilized by , the Mathieu group of degree 11.

·        Objects of type 1 and 3 are stabilized by , the symmetric group of degree 6.

·        Objects of type 2 are stabilized by , the direct product of the cyclic group of order 9 and the dihedral group of order 24.  This stabilizer is of special note because it is the normalizer of a 3-Sylow subgroup of .

 

The following table summarizes the incidence relations between the 5 types of objects of .

 

Type 0

(points of A)

Type 1

(hexads of B and duads of A)

Type 2

(triads of A and triads of B)

Type 3

(hexads of A and duads of B)

Type 4

(points of B)

Type 0

Inclusion

Inclusion

Inclusion

Inclusion

Always

Type 1

Inclusion

Inclusion

5 points in common

4 points in common

Inclusion

Type 2

Inclusion

5 points in common

Inclusion

5 points in common

Inclusion

Type 3

Inclusion

4 points in common

5 points in common

Inclusion

Inclusion

Type 4

Always

Inclusion

Inclusion

Inclusion

Inclusion

 

 

Proposition

With incidence defined as above, our set  is thus a geometry.  Moreover, given any nonmaximal flag, we may extend it to a maximal flag.

Proof

            It is clear from the above table that incidence is symmetric and reflexive.  Since there are obvious dualities in our geometry, the number of cases of nonmaximal flags can essentially be halved, and the proof that any one can be extended to a maximal flag is quite similar to the following specific case.  Given some nonmaximal flag , such that each  is an object of type i, it is clear that  and .  We need to find a point  in B which is contained in .  Since  is composed of a duad from B and a hexad from A, there are two points of B in may be chosen as either one of these, so that any nonmaximal flag F as above may always be extended to a maximal flag.  [Leemans 276]

 

Proposition

The group  acts flag-transitively on .

 

Proof

Take two maximal flags  and , where  and  are objects of type i for each i = 0, 1, 2, 3, 4.  We must find an element  such that .  Since the stabilizer  of objects of type 2 is a subgroup of , we may assume that .  The element  is the union of a triad  in A and a triad  in B.  Recall that  represents three lines in the affine plane, and is thus comprised of 3 duads from B, which implies that there are 3 elements of type 3 which are incident to .  Thus  also acts transitively on duads in B and we have .  The stabilizer of  is a subgroup which is isomorphic to the , which we will denote with .  There are three pairs of points in , and thus three elements of type 1 which are incident to the flag  acts transitively on these three duads, so that we may assume , and the stabilizer of the flag  is a subgroup isomorphic to , which we will denote by .  Since  contains a duad from A, there are 2 objects of type 0 which are incident to the flag , and  is acting transitively on this pair, we know that .  The stabilizer of the flag is a subgroup which is isomorphic to ,  and which we will denote

Finally, there are two elements of type 4 which are incident to the flag , and  acts transitively on them, so that we have .  Thus, there exists , with an element g such that , and  acts flag-transitively on .  [Leemans 277]

 

Proposition

.

 

Proof

If we consider only the elements of types 1, 2, and 4, then  is the automorphism group of the remaining geometry, so that .  We have just shown that the stabilizers of  are all subgroups of , so that .  Therefore, . [Leemans 276]

 

Proposition

Let I = {0, 1, 2, 3, 4}. Then for any , define  to be the subgroup of  which stabilizes the set of objects of types I, where i is in J.  The following table shows the stabilizer of every subset of the objects in .  The proof of this Proposition is given in Dehon et al (in preparation).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Definition

            Given a geometry Γ,  is said to act weakly primitively on Γ if some subgroup , as above, is maximal in  for at least one i in I is said to act residually weakly primitively on Γ if it acts weakly primitively on the residue of every flag of Γ.  [Leemans 274]

 

Proposition

       is residually weakly primitive.

 

Proof

            Obvious (?) by looking at the above table of stabilizers.  [Leemans 277]

 

Definition

            A geometry Γ is said to be connected if any two objects in Γ are either incident with each other, or are mututally incident with another object, or are incident with objects which are mutually incident with another object, et cetera.  Γ is said to be residually connected if the residue of every flag of Γ is connected.  [Pasini 88]

 

Proposition

       is residually connected.

 

Proof

            Obvious from the above inclusion table.  [Leemans 277]

 

Proposition

            The diagram of  is as follows.  [Leemans 276]

 

 

The “C” between types 1 and 2 represents a circular space.  A circular space is a linear space in which any line is incident with exactly 2 points.  The “C*” between types 2 and 3 represents a dual circular space.  A dual circular space represents a circular space in which any two distinct lines meet in a unique point.  [Pasini 14]  For the {1, 2} residue, let objects of type 2 be points, and objects of type 1 be lines.  Given an object, , of type 1, it will contain a hexad of B, which can be partitioned into exactly 2 triads suitable for the construction of elements of type 2. The first triad will form one object of type 2, say, and the other triad will form another, call it .  Define lines and points for the {2, 3} residue similarly. 

To see that the {3, 4} residue is a projective space, consider a flag  of objects of types 0, 1, and 2.  There are three duads of points of B contained in , and thus incident to F.  There are also 3 individual points in B which are incident to F, so that the residue of F is a triangle.  Similarly, the {0, 1} residue is also a triangle.  [Leemans 277]

 

Definition

            A partial linear space is a geometry in which any two points are incident with a unique line.  [Pasini 14]

 

Definition

            A generalized digon is a rank 2 geometry in which any object can be connected to any other object via incidence with at most 3 intermediate objects.  [Pasini 16]

           

Definition

            (IP)2, the intersection property of rank two is said to hold in a geometry Γ if every residue of rank two of Γ is either a partial linear space or a generalized digon.  [Leemans 274]

 

Definition

            A geometry Γ is said to be (2T)1, or locally two-transitive, if for every residue R of Γ of corank 1, the group induced on R acts 2-transitively.  [Leemans 274]

 

Proposition

       is (IP)2 and (2T)1.

 

Proof

            Looking at the column of residues of corank 1 in the subgroup table, one can see that each of the specified subgroups is at least 2-transitive.  The diagram for  shows that every residue of rank two is in fact a linear space, which is a special partial linear space.

 

Definition

            Given a subset J of the set of types I of a geometry Γ and a flag G, the J-shadow of G, denoted , is the set of flags of type J incident to G.  Given two object types G and F, we say that F separates J from G if every object of type J is connected to objects of type G only if it is connected to objects of type F.  If F separates J from G, then we can use the J-shadow to define an ordering on Γ, i.e. .  [Pasini 124]

 

Definition

            The intersection property is said to hold in a geometry Γ if both of the following are true:

            (Isomorphism property) The shadow operator is an isomorphism.

            (Weak intersection property) The partially ordered set is a semilattice with respect to . [Pasini 144]

 

Definition

            Given two geometries Γ and Γ’ a morphism  is a mapping such that if x and y are incident in Γ, then f(x) and f(y) are incident in .  [Pasini 235]

 

Definition

            Let Γ and Γ’ be geometries over the same set of types I, with rank n.  For a given integer , a type preserving morphism  is said to be an m-cover of Γ if for every flag F of Γ of corank m, the morphism f induces an isomorphism from the residue of F in Γ to the residue of F in f  is a universal cover if, for every m-cover , there exists just one m-cover  such that .  [Pasini 245]

 

Proposition

       does not satisfy the intersection property and is not its own universal cover.

 

Proof

            The {0, 1} truncation of  is the geometry consisting of the points and pairs of points of the Steiner system S(5, 6, 12), which is not (IP)2. By Theorem 3.4 of Leemans (1998), this implies that  cannot satisfy the intersection property.

            It is shown in [Meixner 1998] that the universal cover of  is the double cover ,  on which the group  acts flag transitively.

 

References

Beutelspacher, Albrecht and Ute Rosenbaum.  Projective Geometry, University Press, Cambridge  1998.

 

Carmichael, Robert D.  Groups of Finite Order, Dover Publications, Inc., New York 1937.

 

Leemans, Dmitri.  “On a Rank Five Geometry of Meixner for the Mathieu Group ,” in Geometriae Dedicata, volume 85, pages 273-281.

 

Pasini, Antonio.  Diagram Geometries, Clarendon Press, Oxford 1994.