On the primitivity of Cayley digraphs

We give a necessary and sufficient group theoretic condition for a Cayley digraph to be primitive.     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

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     

On a conjecture of huppert for alternating groups

We prove a character degree property of the alternating groups, recently conjectured by Huppert.     

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.     

An “almost” full embedding of the category of graphs into the category of groups

We construct a functor F:Graphs→Groups which is faithful and “almost” full, in the sense that every nontrivial group homomorphism FX→FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f:X→Y. When F is composed with the Eilenberg-Mac Lane space construction K(FX,1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:

(1)

Is every orthogonality class reflective?

(2)

Is every orthogonality class a small-orthogonality class?

have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopěnka's principle and (2) to Vopěnka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopěnka's principle.     

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SLn(R), SU^∗(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO^∗(2n), SO(p,q), SU(p,q) and Sp(p,q). Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with semi-Riemannian metrics.     

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.     

Analogues of Cayley graphs for topological groups

We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogous way to the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth. Supported by project J2245 of the Austrian Science Fund (FWF) and be an IEF Marie Curie Fellowship of the Commission of the European Union.     

Clifford classes for isoclinic groups

Let G be a finite group, N a normal subgroup of G, and an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c Clif(G/N, F) of with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group and character by multiplying each element of each coset of N in G by an appropriate power of a fixed root of unity . We show that there is a simple formula to calculate the Clifford class of in terms of c and . Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups When is a |G/N|-th root of 1, our formula shows that then When = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001     

The largest lengths of conjugacy classes and the Sylow subgroups of finite groups

Let G be a finite nonabelian group, P ∈Syl_p(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/O_p(G)| < bcl(G) in the case where P is abelian. Received: 26 December 2004; revised: 26 January 2005     

Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

In this paper we prove the local existence of complex-valued harmonic morphisms from any compact semisimple Lie group and their non-compact duals. These include all Riemannian symmetric spaces of types II and IV. We produce a variety of concrete harmonic morphisms from the classical compact simple Lie groups SO(n), SU(n), Sp(n) and globally defined solutions on their non-compact duals SO(n,C)/SO(n), SLn(C)/SU(n) and Sp(n,C)/Sp(n).     

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.     

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.     

The snail lemma in a pointed regular category

In this paper we explore the snail lemma in a pointed regular category. In particular, we show that under the presence of cokernels of kernels, the validity of the snail lemma is equivalent to subtractivity of the category. As a corollary, this gives that in the more restrictive context of a normal category the validity of the snail lemma is equivalent to the validity of the snake lemma.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

A note on character kernels in finite groups of prime power order

In this note, we classify the finite groups of prime power order for which all nonlinear irreducible character kernels constitute a chain with respect to inclusion. Received: 6 April 2007