Janko Group Research Articles

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.

Symmetric generation and existence of the Janko group <I>J</I>1

Symmetric generation and existence of the Janko group <I>J</I>1

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.

RX-Complementary generations of the janko groups j1J2And j3

A finite group G with conjugacy class rX is said to be rX-complementary generated if, given an arbitrary x∈G-1, there is a y ∈ rXsuch that G (x,y). The rX-complementary generation of the simple groups was first introduced by Woldar in [17] to show that every sporadic simple group can be generated by an arbitrary element and another suitable element. It is conjectured in [5] that every finite simple group can be generated in this way. In this paper we investigate the rX-complementary generation of the first three Janko groups in an attemp to further develop the techniques of finding rX-complementary generation of the finite simple groups. As a consequence, we obtained all the(p,q,r)-generations of the Janko group J 3, where p,q,r are distinct primes.

(2,3,t)-Generations for the janko group j3

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x l= y m= z n= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J 3. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J 3

A 2-Modular Matrix Group Derived from Steiner Hexads

The 132 Steiner hexads can be grouped in rows to form 11 12 by 12 matrices over F2. Adding the identity to each of these yields 12 involutory matrices, called Steiner matrices, which have properties similar to those of the 12 involutory permutations which are used to construct representations of L(2, 11) and the Janko group J1. The set of rows in the group, GS, generated by these matrices consists of all vectors of weights 1, 5, and 9. There is a surjection of GS onto A12 taking the Steiner matrices onto the involutory permutations, and the kernel is an elementary 2-group. L(2, 11) acts as an automorphism group on GS.

Symmetric Presentations II: The Janko Group J 1

In our paper [6] we exploited parallel behaviour of the Mathieu groups M12 and M24 which we described fully in [4, 5], to obtain presentations of those groups given implicitly in terms of certain highly symmetric generating sets. We made precise the concepts of symmetric generating set, symmetric presentation and primitive presentation, and indicated how these ideas can be generalized to other groups. In the present work, which is a continuation of [6], we turn the process around and seek groups which possess generating sets with certain specified symmetries. We shall find that the most straightforward ways of generalizing [6] immediately give rise to the two smallest Janko groups J1 and J2 Moreover the permutation actions of minimal degree of these two groups, on 266 and 100 letters respectively, emerge naturally and we are readily able to write down our symmetric generators in these representations. All assertions made can be easily checked on a computer, and we are again heavily indebted to the CAYLEY package. That said the final developments of J1 and J2 given here, and in the sequel to this paper, are entirely by hand and are complete in all detail.

The Brauer Tree for J3 in Characteristic 17

We resolve some subtle ambiguities in the placing of the ordinary characters of Janko's group J 3 on the Brauer tree in characteristic 17.

The $5$-modular representations of the Tits simple group in the principal block

In this paper we show how to construct the 5-modular absolutely irreducible representations of the Tits simple group in the principal block, which is the only block of positive defect. Starting with the smallest nontrivial ones, all the others except one pair are obtained as constituents of tensor products of dimension at most 729. The last two we get from a permutation representation of degree 1600. We give an exact description of the construction of the first one of degree 26 by extending its restrictions to several subgroups, a method first used in the existence proof of the Janko group J4 . Using the explicit matrices obtained from the above constructions, we work out the Green correspondents and sources of all the representations and state their socle series.

Efficient presentations for certain simple groups

In this paper we give 2-generator 3-relation presentations for the simple groups PSU(3,5) of order 126000 and the Mathieu group M22Zof order 443520. We also give a 2-generator 2-relation presentation for the Janko group J1of order 175560. This proves that these three simple groups are efficient.

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.

A Characterization and Another Construction of Janko's Group J 3

A Characterization and Another Construction of Janko's Group J 3

Powers of characters of finite groups

Let G be a finite group and θ a complex character of G . Define Irr(θ) to be the set of all irreducible constituents of θ andIrr( G ) to be the set of all irreducible characters of G . The character-covering number of a finite group G , ccn( G ), is defined as the smallest positive integer m such thatIrr(χ m ) =Irr( G ) for allχ∈Irr( G )—{1 G }. If no such positive integer exists we say that the character-covering-number of G is infinite. In this article we show that a finite nontrivial group G has a finite character-covering-number if and only if G is simple and non-abelian and if G is a nonabelian simple group thenccn( G ) ⩽ k 2 − 3 k + 4, where k is the number of conjugacy classes of G . Then we show (using the classification of the finite simple groups) that the only finite group with a character-covering-number equal to two is the smallest Janko's group, J 1 . These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.

A trilinear form for the third Janko group

In this paper we describe the action of the sporadic simple group J 3 on a vector space M of dimension 85 over Q(√−3, √−19). In doing so, we shall prove that J 3 is uniquely determined, up to isomorphism, by the conditions given by Z. Janko [ Ist. Naz. Alta Math. 1 (1968), 25–64] in 1967. We also describe a J 3-invariant symmetric trilinear form ϑ on M. This form, which is uniquely determined up to scalar multiplication, can be used to define a multiplication on A = M ⊕ M ∗ that makes A into a non-associative algebra.

On the Vertices of Irreducible Modules

The same result holds if D is a vertex of an irreducible FG-module Me B. This second assertion generalizes some known results: For p-solvable groups G, it has been shown by Hamernik and Michler [10], that for any irreducible module M in a block B of FG, a vertex of M contains-up to conjugation-the center of a defect group of B. In [13], Michler has shown that all irreducible modules in a block with cyclic defect group A have A as a vertex. A further result in this context was given by Landrock and Michler ([11], Theorem 3.7) as a by-product of their analysis of the principal 2-blocks of the smallest Janko group and the groups of Ree-type. The proof here uses properties of the Green functor to reduce to the case D < G. Then one has to study direct summands of a module induced from a normal subgroup. This is done by inspection of the ring of endomorphisms (Sections 1 and 2). Using these results, the existence of certain B-pairs (see Definition 0.3 below) is shown in Section 3. The results mentioned at the beginning follow then from work done by Brauer and Olsson. The techniques used in the first sections yield a criterion for an ordinary irreducible character X to be of height 0: If F is algebraically closed and M an R-module affording X, then the character X belonging to the p-block B is of height 0 if and only if a vertex of M is a defect group of B and p does not divide the R-rank of a source of M. The proof is given in Section 4.

Block structure of the smallest Janko group

Block structure of the smallest Janko group

Groups with a centralizer of sixth order

Let G be a finite fusion-simple group with a self-centralizing subgroup A of sixth order. It is proved that if the centralizer of the involution from A is an unsolvable subgroup of G of an odd index, then G is isomorphic with the Janko group J1.

FINITE GROUPS WHOSE SYLOW 2-SUBGROUPS HAVE CYCLIC COMMUTATOR SUBGROUPS

The following theorem is proved.THEOREM. Suppose is a finite group such that and . Assume that a Sylow 2-subgroup of is the direct product of subgroups and , where is elementary Abelian and is non-abelian dihedral, semidihedral, or wreathed. Then contains subgroups and with the following properties: 1) ; 2) , and all involutions of are conjugate in ; 3) , and is strongly closed in (with respect to ).As a consequence, a description is given of the finite groups whose Sylow 2-subgroups have cyclic commutator subgroups, the simple ones among which are the following: 1) , where ; 2) and , where is odd; 3) , , the Janko group , and the Ree groups.Bibliography: 12 titles.

CHARACTERIZATION OF SOME FINITE SIMPLE GROUPS BY BIPRIMARY SUBGROUPS. II

In this article, properties of the biprimary subgroups of even order are used to characterize the following series of finite simple groups: the projective special linear groups PSL(2, q), q > 3, the projective special unitary groups PSU(3, 22n), n > 1, the Suzuki groups Sz(q), the Janko group J1 of order 175560, and the Ree-type groups.

Another characterization of the small Janko group

Another characterization of the small Janko group