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Carlo Casolo, Silvio Dolfi
September 19, 2007

### Abstract

We establish linear bounds for the ρ-σ conjecture for irreducible character degrees and conjugacy class sizes of any finite group.

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C Rudloff, A. E Zalesski
September 19, 2007

### Abstract

In this paper we classify irreducible representations of quasi-simple groups with cyclic Sylow p -subgroup P = 〈 g 〉 such that ( g ) has at least one eigenvalue of multiplicity 1.

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Roland Schmidt
September 19, 2007

### Abstract

If L is a lattice, a group is called L -free if its subgroup lattice has no sublattice isomorphic to L . It is easy to see that L 10 , the subgroup lattice of the dihedral group of order 8, is the largest lattice L such that every finite L -free p -group is modular. Therefore in this paper we study finite L 10 -free groups and show that such a group is soluble and (with rather few exceptions) has metacyclic Fitting factor group.

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Andrea Lucchini
September 19, 2007

### Abstract

We obtain some results on the subgroups of finite solvable groups with non-zero Möbius function. These are used to prove a conjecture of Mann in the particular case of prosolvable groups: if G is a finitely generated prosolvable group, then the infinite sum , where H ranges over all open subgroups of G , is absolutely convergent in some right half-plane of the complex plane.

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Martyn R Dixon, Martin J Evans, Howard Smith
September 19, 2007

### Abstract

We show that the class of groups that are soluble-by-(finite rank) is countably recognizable. Also, if every countable subgroup of the group G is (derived length d )-by-(rank r ) then G is (derived length d *)-by-(rank r *) for some bounded d *, r *. Similar results hold for groups that are nilpotent-by-(finite rank).

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Vincenzo Marra, Daniele Mundici
September 19, 2007

### Abstract

Various combinatorial equivalents are given to the Lebesgue state in archimedean lattice-ordered groups with order-unit. The proofs use piecewise linear functions on polyhedra with rational vertices.

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Gregory T Lee, César Polcino Milies, Sudarshan K Sehgal
September 19, 2007

### Abstract

Let K be an infinite field of characteristic different from 2, and G a group containing elements of infinite order. We classify the groups G such that the symmetric units of KG satisfy the identity ( x 1 , x 2 , …, x n ) = 1, for some n .

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Warren Dicks, M. J Dunwoody
September 19, 2007

### Abstract

Let G be a group, let T be an (oriented) G -tree with finite edge stabilizers, and let VT denote the vertex set of T . We show that, for each G -retract V ′ of the G -set VT , there exists a G -tree whose edge stabilizers are finite and whose vertex set is V ′. This fact leads to various new consequences of the almost stability theorem. We also give an example of a group G , a G -tree T and a G -retract V ′ of VT such that no G -tree has vertex set V ′.

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Gilbert Baumslag, Charles F Miller
September 19, 2007

### Abstract

We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel.