Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

On Real Forms of Belyi Surfaces With Symmetric Groups of Automorphisms

In virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study the actions of the symmetric groups S _n on Belyi Riemann surfaces. We show that such surfaces are symmetric and we calculate the number of connected components of the corresponding real forms.     

Complex hyperbolic Kleinian groups with limit set a wild knot

We construct quasi-Fuchsian groups acting on two-dimensional complex hyperbolic space with limit set a wild knot. Also, we study the Teichmüller space T(G) of faithful, discrete, type-preserving representations of a Fuchsian group G of the first kind with parabolic elements in complex hyperbolic space. We show that T(G) is not connected, and that the Toledo invariant does not distinguish different connected components of T(G).     

Applications of a theorem of Singerman about Fuchsian groups

Assume that we have a (compact) Riemann surface S, of genus greater than 2, with , where is the complex unit disc and Γ is a surface Fuchsian group. Let us further consider that S has an automorphism group G in such a way that the orbifold S/G is isomorphic to where is a Fuchsian group such that and has signature σ appearing in the list of non-finitely maximal signatures of Fuchsian groups of Theorems 1 and 2 in [6]. We establish an algebraic condition for G such that if G satisfies such a condition then the group of automorphisms of S is strictly greater than G, i.e., the surface S is more symmetric that we are supposing. In these cases, we establish analytic information on S from topological and algebraic conditions. Received: 4 April 2008     

On cohomologies and extensions of cyclic groups

Cohomology groups Hs(Zn,Zm) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Zn by the Zn-module Zm. Further, for each such a group the number of non-equivalent extensions is determined.     

From local to global similarity of matrix groups

Let G and H be groups of complex n×n matrices. We say that G is an H-like group if every matrix in G is similar to a matrix from H. For several groups H we consider two questions:

(A)

Is every H-like group (simultaneously) similar to a subgroup of H?

(B)

Is H the only H-like group containing H? Among other results we prove that the symmetric group S_n is the only S_n-like group containing S_n.

     

Writing certain commutators as products of cubes in free groups

Let Cu(γ) be the minimal number of cubes required to express an element γ of a free group F. We establish a method for showing that certain equations do not have solutions in free groups. Using it, we find Cu(γ) for certain elements of the derived subgroup of F. If is the wreath product of F by the infinite cyclic group, we also show that every element of W′ is a product of at most one commutator and three cubes in W.     

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in $\text{PSL}\text{(2,}\text{R}\text{)}$) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Abrahamse's interpolation theorem and Fuchsian groups

We generalize Abrahamse's interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalar-valued interpolation theorem for the fixed-point subalgebra of H^∞ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality techniques that parallel Sarason's approach to the interpolation problem for H^∞ [Donald Sarason, Generalized interpolation in H^∞, Trans. Amer. Math. Soc. 127 (1967) 179-203, MR0208383. [26]]. In this process we prove a more general distance formula, very much like Nehari's theorem, and obtain relations between the kernel function for the character automorphic Hardy spaces and the Szegö kernel for the disk. Finally, we examine our interpolation results in the context of the two simplest examples of Fuchsian groups acting on the disk.     

A characterization of Fuchsian groups acting on complex hyperbolic spaces

Let G ? SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is ?-Fuchsian; if G preserves a Lagrangian plane, then G is ?-Fuchsian; G is Fuchsian if G is either ?-Fuchsian or ?-Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that G is conjugate to a subgroup of S(U(1)×U(1, 1)) or SO(2, 1) if each loxodromic element in G is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a ?-Fuchsian group.     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

Morphic groups

A group G is called morphic if every endomorphism α:G→G for which Gα is normal in G satisfies G/Gα≅ker(α). This concept originated in a 1976 paper of Gertrude Ehrlich characterizing when the endomorphism ring of a module is unit regular. The concept has been extensively studied in module and ring theory, and this paper investigates the idea in the category of groups. After developing their basic properties, we characterize the morphic groups among the dihedral groups and the groups whose normal subgroups form a finite chain. We investigate when a direct product of morphic groups is again morphic, prove that a finite nilpotent group is morphic if and only if its Sylow subgroups are morphic, and present some results for the case where a p-group is morphic.     

Small conjugacy classes in the automorphism groups of relatively free groups

Let F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F^′ of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group of the group F/R^′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups F_n/R^′ where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.     

On topologizable and non-topologizable groups

A group G   is called hereditarily non-topologizable if, for every H?G$H?G$, no quotient of H admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked k-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.     

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.     

The symmetric genus of alternating and symmetric groups

The symmetric genus of a finite group G has been defined by Thomas W. Tucker as the smallest genus of all surfaces on which G acts faithfully as a group of automorphisms (some of which may reverse the orientation of the surface). This note announces the symmetric genus of all finite alternating and symmetric groups.     

The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

Almost fixed-point-free automorphisms of soluble groups

Suppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m=4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].     

Eulerian numbers,Newcomb's problem and representations of symmetric groups

Consider a set of n positive integers consisting of μ₁ 1's, μ₂ 2's,…, μ_rr's. If the integer in the ith place in an arrangement σ of this set is σ(i), and a non-rise in σ is defined as σ(i+1)?σ(i), a problem that suggests itself is the determination of the number of arrangements σ with k non-rises. When each μ_i is unity, the problem is that of finding the number A(n, k) of permutations of distinct integers 1, 2,…, n with k descents, a descent being defined as σ(i+1)<σ(i). The number A(n, k) is known as an Eulerian number. The problem of finding the number of arrangements with k non-rises of the more general set, when not all of μ_i are unity, has appeared in the literature as one part of a problem on dealing a pack of cards, this having been proposed by the American astronomer Simon Newcomb (1835–1909).Both the Eulerian numbers and Newcomb's problem have accumulated a substantial literature. The present paper considers these topics from an entirely new stand-point, that of representations of the symmetric group. This approach yields a well-known recurrence for the Eulerian numbers and a known formula for them in terms of Stirling numbers. It also gives the solutions of the Newcomb problem and some recurrences between these solutions, not all of which have been found earlier. A simple connection is found between Stirling numbers and the Kostka numbers of symmetric group representation theory. The Eulerian numbers can also be expressed in terms of the Kostka numbers.The idea which is novel in this treatment and recurs almost as a motif throughout the paper is that of a skew-hook. This occurs in the first place in a very natural way as a picture of the rises and non-rises of σ, with the nodes of the skew-hook labelled successively as σ(1), σ(2),…. As the paper develops, a new form of skew-hook associated with σ emerges. This does not in general depict the rises and non-rises of σ, and it is now the edges, not the nodes, which carry integer labels. A new type of combinatorial number, here called a ψ-function, arises from these edge-labelled skew-hooks. The ψ-functions are intimately related to the Eulerian numbers and the Newcomb solutions and may have further combinatorial applications. The skew-hook treatment casts fresh light on MacMahon's solution of the Newcomb problem and on his “new symmetric functions”, and, if σ(i)?σ(i+1)?s defines an s-descent in σ, on the enumeration of permutations with ks-descents.Also some characters of the symmetric group with interesting properties and recurrences arise in the course of the paper.