A Rank Five Geometry on the Mathieu Group _{ }
MATH 6221
Let _{ }
_{ }
_{ }.
Then _{ }.
_{} is noteworthy because
· It is the only known 5transitive 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 ≤ t ≤ k < 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 3Sylow 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 
With incidence defined as above, our set _{ } is thus a geometry. Moreover, given any nonmaximal flag, we may extend it to a maximal flag.
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]
The group _{ } acts flagtransitively on _{ }.
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 flagtransitively on _{ }. [Leemans 277]
_{}.
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]
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]
_{ } is residually weakly primitive.
Obvious (?) by looking at the above table of stabilizers. [Leemans 277]
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]
_{ } is residually connected.
Obvious from the above inclusion table. [Leemans 277]
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]
A partial linear space is a geometry in which any two points are incident with a unique line. [Pasini 14]
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]
(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]
A geometry Γ is said to be (2T)_{1}, or locally twotransitive, if for every residue R of Γ of corank 1, the group induced on R acts 2transitively. [Leemans 274]
_{ } is (IP)_{2} and (2T)_{1}.
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 2transitive. The diagram for _{ } shows that every residue of rank two is in fact a linear space, which is a special partial linear space.
Given a subset J of the set of types I of a geometry Γ and a flag G, the Jshadow 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 Jshadow to define an ordering on Γ, i.e. _{ }. [Pasini 124]
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]
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]
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 mcover 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 mcover _{ }, there exists just one mcover _{ } such that _{ }. [Pasini 245]
_{ } does not satisfy the intersection property and is not its own universal cover.
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 273281.
Pasini, Antonio. Diagram Geometries, Clarendon Press, Oxford 1994.