Suzuki Group Sz Research Articles

Multiplicities of irreducible components of restrictions of complex representations of finite groups to certain subgroups

Let G be a finite group, n an integer and P a field; we say that G ∈ Rn(P) if for any irreducible representation X of G there exists a proper subgroup H (depending on X) and an irreducible representation ξ of H which is a component of X|H with multiplicity ≤ n. We prove that G ∈ Rn(P) for an algebraically closed field P provided S ∈ Rn(P) for the universal central extensions S of all nonabelian simple quotients of G. We also prove that the universal central extensions of all projective special linear groups PSL(m.,q), of alternating groups Am and Suzuki groups Sz(q) belong to R1(C).

The structure of the projective indecomposable modules of the Suzuki group 𝑆𝑧(8) in characteristic 2

This paper describes the socle series of the projective indecomposable modules and of tensor products of simple modules for the simple group Sz ( 8 ) {\text {Sz}}(8) in characteristic 2. The results have been obtained by computational means and the various steps are described. The main algorithm was modified to allow for parallel execution on a network of workstations. This made possible the effective handling of modules of degree 4030.

Constructions of large translation nets with nonabelian translation groups

In this paper the first infinite series of translation nets with nonabelian translation groups and a large number of parallel classes are constructed. For that purpose we investigate partial congruence partitions (PCPs) with at least one normal component.Two series correspond to partial congruence partitions containing one normal elementary abelian component. The construction results by using some basic facts about the first cohomology group of the translation group G regarded as an extension of the normal component which itself is a group of central translations.The other series correspond to partial congruence partitions containing two normal nonabelian components. The constructions are based on the well known automorphism method which leads to so-called splitting translation nets. By investigating the Suzuki groups Sz(q), the protective unitary groups PSU(3, q 2) and the Ree groups R(q) as doubly transitive permutation groups, we obtain examples of nonabelian groups admitting a large number of pairwise orthogonal fixed-point-free group automorphisms.

Intersection pattern of the classical ovoids in symplectic 3-space of even order

For s = 2 e , e > 1 odd, we determine how the copies of the Suzuki group Sz( s) in the symplectic group Sp(4, s) intersect. Using this information we determine how the classical ovoids in symplectic 3-space W( s) meet and obtain a complete set of double coset representatives of Sz( s) in Sp(4, s). We also note that the permutation representation of Sp(4, s) on the cosets of Sz( s) is multiplicity free, and its irreducible constituents are explicitly determined. Indeed, we show that the complex Hecke algebra of this permutation representation is isomorphic to the center of the complex group algebra of Sz( s). A combinatorial offshoot of this study is the construction of several new series of Buekenhout diagram geometries of type ▪ which are embedded as subgeometries of miquelian and Suzuki-Tits inversive planes.

The 2-transitive ovoids

An ovoid 0 in a classical polar space is a set of singular points such that every maximal totally singular subspace contains just one point in Lo. Ovoids are intimately connected with other combinatorial objects, including translation planes, spreads, partial geometries, codes, generalized hexagons, and Kerdock sets (see [S, 9, 163 for example). Interestingly enough, many of the known ovoids are in fact 2-transitive-that is, they admit a 2-transitive automorphism group (the notion of the automorphism group is made precise in (2.3)). For example, the Suzuki groups Sz(q) and the Ree groups *G,(q) act 2-transitively on ovoids in 4-dimensional symplectic geometry and 7-dimensional orthogonal geometry, respectively. Furthermore, the unitary groups PSU,(q) (for suitable prime powers q) and the linear groups PSL,(q3) (with q even) act 2-transitively on ovoids in 7or 8-dimensional orthogonal geometry. The occurrence of such a large number of 2-transitive ovoids suggests that a classification of them is worthwhile, much in the same spirit as Kantor’s classification of the finite linear spaces whose automorphism group acts 2-transitively on points [lo]. The classification of the 2-transitive ovoids appears as our Main Theorem in Section 2. We discover no new ovoids, however we obtain some new results concerning the number of isomorphism classes of 2-transitive ovoids (see Section 2). Our proof relies on the classification of the finite 2-transitive permutation groups, which in turn relies on the recent classification of finite simple groups. WC also draw upon several facts from the modular representation theory of finite groups.

Is J 1 a Subgroup of the Monster?

Bulletin of the London Mathematical SocietyVolume 18, Issue 4 p. 349-350 Notes and papers Is J1 a Subgroup of the Monster? Robert A. Wilson, Robert A. Wilson Department of Pure Mathematics and Mathematical Statistics University of Cambridge, 16 Mill Lane Cambridge CB2 1SBSearch for more papers by this author Robert A. Wilson, Robert A. Wilson Department of Pure Mathematics and Mathematical Statistics University of Cambridge, 16 Mill Lane Cambridge CB2 1SBSearch for more papers by this author First published: July 1986 https://doi.org/10.1112/blms/18.4.349Citations: 5AboutRelatedInformationPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessClose modalShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL No abstract is available for this article.Citing Literature Volume18, Issue4July 1986Pages 349-350 RelatedInformation RecommendedIs the Suzuki group Sz(8) a subgroup of the Monster?Robert A. Wilson, Bulletin of the London Mathematical SocietyStructure of the cuspidal rational torsion subgroup of J1(pn)Yifan Yang, Jeng-Daw Yu, Journal of the London Mathematical SocietySubgroup Families Controlling p-Local Finite GroupsCarles Broto, Natàlia Castellana, Jesper Grodal, Ran Levi, Bob Oliver, Proceedings of the London Mathematical SocietyPSL2(59) is a Subgroup of the MonsterPetra E. Holmes, Robert A. Wilson, Journal of the London Mathematical SocietyThe uniqueness of PSU 3(8) in the MonsterRobert A. Wilson, Bulletin of the London Mathematical Society

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.

On 2-Modular Representations of the Suzuki Groups

In this paper we apply methods of G. Higman, outlined in [2], to determine on which 2-groups Q the Suzuki groups Sz(q) can act as a group of automorphisms with an element of order 5 acting fixed point freely. We find that Q is elementary abelian and we determine precisely the action of Sz(q) on Q. To state what this action is, we need some definitions. In his paper [3], Suzuki defines the group G(q), here denoted by Sz(q), as generated as a subgroup of GL(4,q) by a certain collection of matrices. We let M be a 4-dimensional vector space over GF(q) affording this representation of Sz(q). If N is a module over the group-ring GF (q) Sz(q) (here GF(q) is the field of q elements), then we say that the action of Sz(q) on N is natural if N is isomorphic as GF (q) -Sz(q) module to a direct sum of copies of M1. If N is an elementary abelian 2-group admitting Sz(q) as a group of operators, then we say that the action of Sz(q) on N is natural if N is isomorphic, as GF (2) Sz (q) module to a GF (q) Sz (q) module on which Sz (q) acts naturally. With this notation, the following is a precise statement of the main result of this paper.

CHARACTERIZATION OF SOME FINITE SIMPLE GROUPS

The paper contains a characterization, by means of their biprimary subgroups of even order, of the following finite simple groups: the special projective linear groups PSL(2, q) (q > 3), the Suzuki groups Sz(q), and Janko's group J of order 175,560.