Unable to retrieve citations for this document

Retrieving citations for document...

Zvonimir Janko
February 12, 2007

### Abstract

We determine here the structure of the groups of the title which are of exponent at least 4. It turns out that such a group is either cyclic or of maximal class (i.e., dihedral, semidihedral or generalized quaternion) or a uniquely determined group of order 2 5 (Theorem 1.1). This solves Problem no. 523 in Berkovich [Y. Berkovich. Groups of prime power order . In preparation.] for p = 2. The corresponding problem for p > 2 is open but very difficult.

Unable to retrieve citations for this document

Retrieving citations for document...

Jon F Carlson
February 12, 2007

Cite
Requires Authentication
Accessible
PDF
Unable to retrieve citations for this document

Retrieving citations for document...

Morton E Harris
February 12, 2007

### Abstract

Let G be a finite p -solvable group for some prime p and let A be a subgroup of Aut( G ) with order relatively prime to G , such that A stabilizes a p -block b of G and acts trivially on a defect group P of b . All ordinary characters in b are A -stable by a result of Watanabe. Also there is a block w ( b ) of C G ( A ) such that the Glauberman–Isaacs correspondent characters comprise the ordinary irreducible characters of w ( b ), and there is a Morita equivalence, and hence a perfect isometry, between the blocks b and w ( b ), given by a bimodule M with vertex Δ P and an endo-permutation module as source and which induces the Glauberman–Isaacs correspondence on the character level. This gives a different proof of results of several authors.

Unable to retrieve citations for this document

Retrieving citations for document...

F. G Timmesfeld
February 12, 2007

Cite
Requires Authentication
Accessible
PDF
Unable to retrieve citations for this document

Retrieving citations for document...

Nikolay Nikolov
February 12, 2007

### Abstract

We prove that every quasisimple group of classical type is a product of boundedly many conjugates of a quasisimple subgroup of type A n .

Unable to retrieve citations for this document

Retrieving citations for document...

Alexander J. E Ryba
February 12, 2007

Cite
Requires Authentication
Accessible
PDF
Unable to retrieve citations for this document

Retrieving citations for document...

G. Y Chen, V. D Mazurov, W. J Shi, A. V Vasil'ev, A. Kh Zhurtov
February 12, 2007

### Abstract

1 Introduction For a finite group G , denote by ω( G ) the spectrum of G , i.e., the set of orders of elements in G . This set is closed under divisibility and hence is uniquely determined by the subset μ( G ) of elements in ω( G ) which are maximal under the divisibility relation. A group G is said to be recognizable by ω( G ) (for short, recognizable ) if every finite group H with ω( H ) = ω( G ) is isomorphic to G . In other words, G is recognizable if h ( G ) = 1 where h ( G ) is the number of pairwise non-isomorphic groups H with ω( H ) = ω( G ). It is known that h ( G ) = ∞ for every group G that has a non-trivial soluble normal subgroup, and so the recognizability problem is interesting only for groups with trivial soluble radical, and first of all for simple and almost simple groups. The goal of this paper is to resolve the recognizability problem for the groups PGL 2 ( q ), i.e., to find h (PGL 2 ( q )) for all q .

Unable to retrieve citations for this document

Retrieving citations for document...

Gilbert Baumslag, Benjamin Fine, Anthony M Gaglione, Dennis Spellman
February 12, 2007

### Abstract

Here we continue the study of discriminating groups as introduced by Baumslag, Myasnikov and Remeslennikov in [G. Baumslag, A. G. Myasnikov and V. N. Remeslennikov. Discriminating and codiscriminating groups. J. Group Theory 3 (2000), 467–479.]. First we give examples of finitely generated groups which are discriminating but not trivially discriminating, in the sense that they do not embed their direct squares, and then we show how to generalize these examples. In the opposite direction we show that if F is a non-abelian free group and R is a normal subgroup of F such that F / R is torsion-free, then F / R ′ is non-discriminating.

Unable to retrieve citations for this document

Retrieving citations for document...

Gerald Williams
February 12, 2007

### Abstract

A generalized triangle group is a group that can be presented in the form where l , m , n ∈ {2, 3, 4, …} ∪ {∞} and w ( x , y ) is an element of the free product involving both x and y . A homomorphism : Γ → G is said to be essential if have orders l , m , n respectively. Every generalized triangle group Γ admits an essential representation to PSL(2, ). In most cases there will be such a representation with infinite or non-elementary image. Vinberg and Kaplinsky say that Γ is pseudo-finite if the image of any essential representation Γ → PSL(2, ) is finite and they have obtained a partial classification of such groups. Extending this concept, we call Γ pseudo-elementary if the image of any essential representation Γ → PSL(2, ) is elementary. In this paper we classify the pseudo-elementary generalized triangle groups Γ with n ≥ 3 and obtain partial results in the case n = 2.

Unable to retrieve citations for this document

Retrieving citations for document...

Susan Hermiller, Zoran Šunić
February 12, 2007

### Abstract

We show that every right-angled Artin group A Γ defined by a graph Γ of finite chromatic number is poly-free with poly-free length bounded between the clique number and the chromatic number of Γ. Further, a characterization of all right-angled Artin groups of poly-free length 2 is given, namely the group A Γ has poly-free length 2 if and only if there exists an independent set of vertices D in Γ such that every cycle in Γ meets D at least twice. Finally, it is shown that A Γ is a semidirect product of two free groups of finite rank if and only if Γ is a finite tree or a finite complete bipartite graph. All of the proofs of the existence of polyfree structures are constructive.