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Serge Bouc, Jacques Thévenaz
March 11, 2008

### Abstract

Let k be a field of characteristic p , let P be a finite p -group, where p is an odd prime, and let D ( P ) be the Dade group of endo-permutation kP -modules. It is known that D ( P ) is detected via deflation–restriction by the family of all sections of P which are elementary abelian of rank ≤ 2. In this paper, we improve this result by characterizing D ( P ) as the limit (with respect to deflation–restriction maps and conjugation maps) of all groups D ( T/S ) where T/S runs through all sections of P which are either elementary abelian of rank ≤ 3 or extraspecial of order p 3 and exponent p .

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John Bamberg, Geoffrey Pearce, Cheryl E Praeger
March 11, 2008

### Abstract

A transitive decomposition is a pair (Γ, 𝒫) where Γ is a graph and 𝒫 is a partition of the arc set of Γ, such that there exists a group of automorphisms of Γ which leaves 𝒫 invariant and transitively permutes the parts in 𝒫. This paper concerns transitive decompositions where the group is a primitive rank 3 group of ‘grid’ type. The graphs Γ in this case are either products or Cartesian products of complete graphs. We first give some generic constructions for transitive decompositions of products and Cartesian products of copies of an arbitrary graph, and we then prove (except in a small number of cases) that all transitive decompositions with respect to a rank 3 group of grid type can be characterized using these constructions. Furthermore, the main results of this work provide a new proof and insight into the classification of rank 3 partial linear spaces of product action type studied by Devillers.

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A. O Asar
March 11, 2008

### Abstract

This work continues the study of the subgroups of totally imprimitive groups of finitary permutations which was started in [A. O. Asar. On finitary permutation groups. Turkish J. Math. 30 (2006), 101–116]. Here some characterizations of non- FC -subgroups and subnormal subgroups of stabilizers of these groups are obtained, and the non-triviality of the core of G in the normalizers of blocks is shown.

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Shikun Ou, Dengyin Wang
March 11, 2008

### Abstract

In this note, we determine all automorphisms of the standard Borel subgroup of the symplectic group Sp(2 m, R ), when m ≥ 3 and R is a commutative ring with identity in which 2, 3 are invertible.

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Jairo Z Gonçalves, Ángel del Río
March 11, 2008

### Abstract

Let G be a finite group and ℤ G its integral group ring. We show that if α is a nontrivial bicyclic unit of ℤ G , then there are bicyclic units β and γ of different types, such that 〈α, β〉 and 〈α, γ〉 are non-abelian free groups. In the case when G is non-abelian of order coprime to 6 we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ℤ G , such that 〈 u m , v 〉 is a free non-abelian group for all sufficiently large positive integers m .

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Heather Armstrong, Bradley Forrest, Karen Vogtmann
March 11, 2008

### Abstract

We study the action of the group Aut( F n ) of automorphisms of a finitely generated free group on the degree 2 subcomplex of the spine of Auter space. Hatcher and Vogtmann showed that this subcomplex is simply connected, and we use the method described by K. S. Brown to deduce a new presentation of Aut( F n ).

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Eric M Freden, Jennifer Schofield
March 11, 2008

### Abstract

We generalize an example of Higman to a family of groups H m , m = 1, 2, 3, …, each of which is isomorphic to the amalgam of a solvable Baumslag–Solitar group with itself. We explicitly compute the growth function of H 3 . This function is algebraic but not rational. The construction uses the tree-like geometry of the Cayley graph and ideas from formal language theory. Modification of the proof allows the computation of the growth function of H m for odd m .