A Rank
Five Geometry on the Mathieu Group
MATH 6221
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 ≤ 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 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 |
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
flag-transitively 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
flag-transitively 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).
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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 two-transitive, if for every residue R of Γ of corank 1, the group induced on R acts 2-transitively. [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
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.
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]
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 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]
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 273-281.
Pasini, Antonio. Diagram Geometries, Clarendon Press, Oxford 1994.
