Hall-Janko Group Research Articles

A differential equation with monodromy group 2.J2

We construct a sixth order differential equation having the central extension of C2 by the Hall–Janko group J2 as monodromy group. Moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation.

Sporadic simple groups and quotient singularities

We show that if a faithful irreducible representation of a central extension of a sporadic simple group with centre contained in the commutator subgroup gives rise to an exceptional (resp. weakly exceptional but not exceptional) quotient singularity, then that simple group is the Hall–Janko group (resp. the Suzuki group).

Six-dimensional exceptional quotient singularities

We classify six-dimensional exceptional quotient singularities and show that seven- dimensional exceptional quotient singularities do not exist. Inter alia we prove that the irre- ducible six-dimensional projective representation of the sporadic simple Hall-Janko group gives rise to an exceptional quotient singularity. where Ei is a ξ-exceptional divisor, and bi ∈ Q. Let B be an effective Q-divisor on V. Put B = m X i=1 aiBi, where Bi is a prime Weil divisor on V , and ai ∈ Q>0. Suppose that B is a Q-Cartier divisor. Then m X i=1 ai ¯ Bi ∼Q ξ ∗ m X i=1 aiBi !

Some self-dual codes invariant under the Hall–Janko group

We construct self-dual codes of length 100 invariant under the Hall–Janko group J 2 . We also construct a 2- ( 100 , 22 , 168 ) design with the automorphism group J 2 : 2 , which is the extension of J 2 by its outer automorphism.

Double circulant codes from two class association schemes

Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary codes attached to rank three groups.

BROUÉ'S CONJECTURE FOR THE HALL–JANKO GROUP AND ITS DOUBLE COVER

Broué's abelian defect conjecture suggests a deep link between the module categories of a block of a group algebra and its Brauer correspondent, viz. that they should be derived equivalent. We are able to verify Broué's conjecture for the Hall–Janko group, even its double cover $2.J_2$, as well as for $U_3(4)$ and ${\rm Sp}_4(4)$. In fact we verify Rickard's refinement to Broué's conjecture and show that the derived equivalence can be chosen to be a splendid equivalence for these examples.2000 Mathematical Subject Classification: 20C20, 20C34.

On a rank four geometry for the Hall–Janko sporadic group

Starting from the well-known rank three graph of degree 63 on 100 points associated to the Hall–Janko group J 2, we construct a rank four residually connected and firm geometry Γ on which J 2 acts flag-transitively and residually weakly primitively (RWPRI). It is the first rank four geometry satisfying RWPRI that is currently known for J 2 and it shows that the RWPRI rank of J 2 is at least four. The diagram of Γ is the following:

Hyperbolic Plane Reflections and the Hall–Janko Group

We introduce the notion of a hyperbolic plane reflection in symplectic space over a finite field of characteristic 3 and show that the group 2HJ, where HJ is the Hall–Janko simple group, is generated by a set of 315 hyperbolic plane reflections in symplectic 6-space.

A (c. L*-Geometry for the Sporadic Group J2

We prove the existence of a rank three geometry admitting the Hall–Janko group J2 as flag-transitive automorphism group and Aut(J2) as full automorphism group. This geometry belongs to the diagram (c·L*) and its nontrivial residues are complete graphs of size 10 and dual Hermitian unitals of order 3.

Modular representations of the hall-janko group

We study modular representations of the Hall-Janko group and its double cover in characteristics 2, 3 and 5. In particular, we determine all extensions of simple modules. Results on the group G 2(2) ≅ PSU(3,3), which is isomorphic to a maximal subgroup of the Hall-Janko group, are also included.

Classifying Geometries with Cayley

In 1961, J.Tits has described a way to define a geometry from a group and a collection of subgroups. A lot of interesting geometrical objects arising from this definition have been studied by many geometers. The problem of finding all the geometries associated to a given group has been solved by hand, only for very small groups. A partial classification of the geometries of the Hall-Janko group has been recently obtained by M.Hermand, with the help of CAYLEY. Here we present a set of CAYLEY programmes to classify all the primitive, firm, residually connected and flag-transitive geometries associated to a given group G . As an application, we give the results obtained for the group W (E6).

Strongly irreducible collineation groups and the Hall-Janko group

Strongly irreducible collineation groups and the Hall-Janko group

On the geometry of the Hall-Janko group on 315 points

On the geometry of the Hall-Janko group on 315 points

A New Lattice for the Hall-Janko Group

A New Lattice for the Hall-Janko Group

A new lattice for the Hall-Janko group

Six by six matrix generators with entries in the rational numbers adjoined by a fifth root of unity, and most of these entries 0, are given for the proper central extension of the cyclic group of order 2 by the Hall-Janko group. They are found to preserve a 24 24 -dimensional lattice giving a packing of unit spheres with density about one fourth that of the Leach lattice.

On certain permutation representations of the Hall–Janko group

SynopsisCertain permutation representations of the Hall–Janko group J2 are studied. These representations are of interest in connection with the problem of whether J2 can act as astrongly irreducible collineation group of a finite projective plane.

The brauer characters of the hall-janko group

(1988). The brauer characters of the hall-janko group. Communications in Algebra: Vol. 16, No. 2, pp. 357-398.

The geometry of the Hall-Janko group as a quaternionic reflection group

The geometry of the Hall-Janko group as a quaternionic reflection group

Characterization of groups of Janko-Ree type and the Hall-Janko group

Characterization of groups of Janko-Ree type and the Hall-Janko group

Finite groups in which a Sylow two-subgroup of the centralizer of some involution is of order 16

It is proved that the sectional two-rank of a finite group G having no subgroup of index two is at most four if a Sylow two-subgroup of the centralizer of some involution of G is of order 16. This implies the following assertion: If G is a finite simple group whose order is divisible by 25 and the order of the centralizer of some involution of G is not divisible by 25, then G is isomorphic to the Mathieu group M12 or the Hall-Janko group J2.