Janko Group Research Articles

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).

A rank 3 geometry for the O’Nan group connected with the Livingstone graph

We construct a rank 3 geometry ( O 0 N) over the diagram a a a 1 10 1 c 8 5 8 whose automorphism group is the O’Nan sporadic simple group. The maximal parabolic subgroups are the Janko group J1, 2 × S5 and the Mathieu group M11. Our construction is based on a convenient amalgam of known geometries of rank 2 for J1 and M11 extracted from the subgroup lattice of O 0 N.

Regular 4-polytopes from the Livingstone graph of Janko’s first group

The Janko group J 1 has, up to duality, exactly two regular rank four polytopes, of respective Schlafli types {5,3,5} and {5,6,5}. The aim of this paper is to give geometric constructions of these two polytopes, starting from the Livingstone graph.

Torsion units in integral group rings of Janko simple groups

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.

Decomposition numbers of non-principal blocks of [formula omitted] for characteristic 3

In this paper, the decomposition numbers of the non-principal blocks of the largest Janko group J 4 [Z. Janko, A new finite simple group of order 86,775,571,046,077,562,880 which possesses M 24 and the full covering group of M 22 as subgroups, J. Algebra 42 (1976) 564-596] for characteristic 3 are determined up to some unknown parameters. We are also concerned with the decomposition numbers of maximal 2-local subgroups of J 4 in odd characteristics.

The admissibility of sporadic simple groups

In 1955 Hall and Paige conjectured that a finite group is admissible, i.e., admits complete mappings, if its Sylow 2-subgroup is trivial or noncyclic. In a recent paper, Wilcox proved that any minimal counterexample to this conjecture must be simple, and further, must be either the Tits group or a sporadic simple group. In this paper we improve on this result by proving that the fourth Janko group is the only possible minimal counterexample to this conjecture: John Bray reports having proved that this group is also not a counterexample, thus completing a proof of the Hall–Paige conjecture.

On the Ranks of Janko Groups J 1, J 2, J 3 and J 4

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji , where i = 1, 2, 3, 4.

On locally spherical polytopes of type {5, 3, 5}

There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is J 1 × J 1 × L 2 ( 19 ) , where J 1 is the first Janko group, of order 175560, and L 2 ( 19 ) is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one.

Symmetric presentation of the Janko group J4

We demonstrate how readily a definition of the largest Janko group J4 follows from a primitive action of the Mathieu group M24 by exhibiting J4 as an image of the progenitor 2*3795:M24. This symmetric presentation is converted into an ordinary presentation on three generators.

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.

O'NAN GROUP UNIQUELY DETERMINED BY THE CENTRALIZER OF A 2-CENTRAL INVOLUTION

In this paper we give a self-contained existence and uniqueness proof for the sporadic O'Nan group ON by showing that it is uniquely determined up to isomorphism by the centralizer H of a 2-central involution z. We establish for such a simple group G a presentation in terms of generators and defining relations and a faithful permutation representation of degree 2.624.832 with a uniquely determined stabilizer isomorphic to the small sporadic Janko group J1. We also calculate its character table by new methods and determine a system of representatives of the conjugacy classes of G.

Block designs and strongly regular graphs constructed from the group U(3,4)

We show a construction of the projective plane P G(2; 16) and the Hermitian unital S(2; 5; 65) from the unitary group U(3; 4). Fur- ther, we construct two block designs, a 2-(65; 15; 21) design and a 2- (65; 26; 250) design, and two strongly regular graphs with parameters (208; 75; 30; 25) and (416; 100; 36; 20). These incidence structures are de- ned on the elements of the conjugacy classes of the maximal subgroups of U(3; 4). The group U(3; 4) acts transitively as an automorphism group of the so constructed designs and strongly regular graphs. The strongly regular graph with parameters (416; 100; 36; 20) has the full automorphism group of order 503193600, isomorphic to G(2; 4) : Z2. Since the Janko group J2 is a subgroup of G(2; 4), J2 acts as an automorphism group of the constructed SRG(416; 100; 36; 20).

Symmetric generation and existence of [formula omitted], the automorphism group of the third Janko group

We make use of the primitive permutation action of degree 120 of the automorphism group of L 2 ( 16 ) , the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J 3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J 3 : 2 , to obtain the order of J 3 and to prove its simplicity. A presentation for J 3 : 2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J 3 : 2 .

On the thin regular geometries of rank four for the Janko group J1

On the thin regular geometries of rank four for the Janko group J1

The Residually Weakly Primitive Geometries of J 3

We summarize the classification of all firm and residually connected geometries satisfying the conditions (IP)2 and (2T)1 and on which the Janko group J 3 acts flag-transitively and residually weakly primitively. We state some facts regarding the results. The complete list of geometries is available as a supplement to this note [Leemans 03c).

An Algorithmic Analysis of the Intersection Property

AbstractIn this paper, efficient algorithms are given to test the intersection property and some of its variations on flag-transitive coset geometries. These algorithms are then applied to geometries of some sporadic groups, namely the Mathieu groups M11, M12, M22and M23, the Janko groups J1, J2and J3 and the Higman-Sims group HS.

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:

A characterization of the Janko group $J_1$ by an active fragment of its character table

A characterization of the Janko group $J_1$ by an active fragment of its character table

The cohomology of the Mathieu group M23

and, as a result of our calculation Hi(M23;Z) = 0 for i < 5. In particular M23 is the first known counterexample to the conjecture that if G is a finite group with Hi(G;Z) = 0, i = 1, 2, 3, then G = {1}‡. (M11, the first Mathieu group and J1, the first Janko group also satisfy Out(G) = Mult(G) = 1, but for both of these groups H3(G;Z) = 0.) It would be tempting to amend the conjecture. It is very likely that it only fails for a very small number of the sporadics among the simple groups. So one might well suspect that there is a (small) finite number n so that H(G;Z) = 0 for 0 < i ≤ n implies that G = {1} if G is finite. But I have no idea as to a suitable candidate for n.