On fixed point sets and Lefschetz modules for sporadic simple groups

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p³, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.     

Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A$A$ and B$B$ are nontrivial conjugacy classes, then AB$AB$ is not a conjugacy class. We also prove that if G$G$ is a finite simple group of Lie type and A$A$ and B$B$ are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB$AB$ is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p$p$-elements with p≥5$p\ge 5$.     

The 3-modular decomposition numbers of the sporadic simple suzuki group

The purpose of this paper is to determine the 3-modular decomposition numbers for the sporadic simple Suzuki group Suz. The results are obtained by a combined use of character theory, explicit construction of modules, and condensation techniques with help of the computer algebra systems MOC, MeatAxe, Tensorcondense, and GAP.     

On simple modules and normal subgroups of p-solvable groups

Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Next, denote by IRR⁰(FG|M) the set of all simple FG-modules V lying over M such that the p-part of dim_F V is as small as possible. In this paper, |IRR⁰(FG|M)| and the vertices of modules in IRR⁰(FG|M) are determined. The p-blocks of G to which modules in IRR⁰(FG|M) belong are also determined.Received: 5 December 2003     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

Broué's conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of N_G(D) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O'Nan simple group and the Higman-Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.     

Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the smallest extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group. Received: 18 February 2002     

Schur indices of characters of groups related to finite sporadic simple groups

LetG be a finite sporadic simple group. Then there exist groupsn.G., n.G.2 and, in casen is even,n.G.2i, the group isoclinic to but not isomorphic ton.G.2. The Schur indices of all irreducible characters of these groups are computed. In a previous paper this was done for the groupsn.G (with one exception). The division algebra corresponding to a character is determined by all the local Schur indices. These are all listed in the tables in Section 6 using the notation from the ATLAS.     

On the Navarro-Willems conjecture for blocks of finite groups

We prove that a set of characters of a finite group can only be the set of characters for principal blocks of the group at two different primes when the primes do not divide the group order. This confirms a conjecture of Navarro and Willems in the case of principal blocks.     

Counting characters of small degree in upper unitriangular groups

Let U_n denote the group of upper n×n unitriangular matrices over a fixed finite field of order q. That is, U_n consists of upper triangular n×n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U_n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U_n of degree q^e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1,2 and 3, and thus show that the conjecture is true in those cases.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Lie symmetry of the Landau-Lifshitz-Gilbert equation and exact linearization in the Minkowski space

In this paper we present a linear representation of the Landau-Lifshitz-Gilbert equation for describing the magnetization of ferromagnetic materials. According to Lies theory, we prove that this equation admits a superposition principle and its formula is derived. The underlying vector space of the Landau-Lifshitz-Gilbert equation is found to be a projective Minkowski space denoted by of which the projective proper orthochronous Lorentz group PSO ^o(3,1) left acts. By the Lie symmetry a group preserving scheme is developed, which improves the computational accuracy and efficiency.     

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SLn(q) in the non-defining characteristic ?, describe restrictions of irreducible modules from GLn(q) to SLn(q), classify complex irreducible characters of SLn(q) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SLn(q).     

Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups

This paper solves for torsion free and torsion abelian groups G the problem of presenting nth powers Δⁿ(G) of the augmentation ideal Δ(G) of an integral group ring , in terms of the standard additive generators of Δⁿ(G). A concrete basis for Δⁿ(G) is obtained when G itself has a basis and is torsion. The results are applied to describe the homology of the sequence Δn(N)G?Δn(G)?Δn(G/N).     

A new approach to center conditions for simple analytic monodromic singularities

In this paper we present an alternative algorithm for computing Poincaré-Lyapunov constants of simple monodromic singularities of planar analytic vector fields based on the concept of inverse integrating factor. Simple monodromic singular points are those for which after performing the first (generalized) polar blow-up, there appear no singular points. In other words, the associated Poincaré return map is analytic. An improvement of the method determines a priori the minimum number of Poincaré-Lyapunov constants which must cancel to ensure that the monodromic singularity is in fact a center when the explicit Laurent series of an inverse integrating factor is known in (generalized) polar coordinates. Several examples show the usefulness of the method.