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Anton Evseev, George Wellen
November 6, 2008

### Abstract

We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalize this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small dimensions may be calculated.

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Siddhartha Sarkar
November 6, 2008

### Abstract

Let G be a finite group. A non-negative integer g is called a genus of G if G acts faithfully on a compact orientable surface Σ g preserving orientation. The set of all such possible genera g ⩾ 2 for a finite group G is called the genus spectrum of G ; after re-scaling it is called the reduced genus spectrum for G . The reduced genus spectrum of a given finite group G contains all sufficiently large numbers. We will describe the reduced genus spectrum of finite p -groups of exponent p , where p is a prime, and also for p -groups of maximal class with order less than or equal to p p .

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Markus Linckelmann, Nadia Mazza
November 6, 2008

### Abstract

We define the notion of the Dade group of a fusion system and show that some of the gluing and detection results for Dade groups of finite p -groups due to Bouc and Thévenaz in [S. Bouc and J. Thévenaz. Gluing torsion endo-permutation modules. (Preprint.)], [S. Bouc and J. Thévenaz. A sectional characterization of the Dade group. J. Group Theory 11 (2008), 155–183.] extend to Dade groups of fusion systems.

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Michael Geline
November 6, 2008

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Andrei Marcus
November 6, 2008

### Abstract

We show that G -graded Rickard equivalences defined over small fields preserve Clifford classes associated to characters. These equivalences are compatible with operation on Clifford classes defined in terms of central simple crossed products.

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Michael Giudici, Shane Kelly
November 6, 2008

### Abstract

A finite transitive permutation group is said to be elusive if it has no fixed-point free elements of prime order. In this paper we show that all elusive groups G = N ⋊ G 1 with N an elementary abelian minimal normal subgroup and G 1 cyclic, can be constructed from transitive subgroups of AGL(1, p 2 ), for p a Mersenne prime, acting on the set of p ( p + 1) lines of the affine plane AG(2, p ).

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Cheryl E. Praeger, Jing Xu
November 6, 2008

### Abstract

We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n , there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of Z ⋊ H . We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than Z ⋊ H . As an application we construct, for each prime power q , a digraph with q – 1 vertices and automorphism group equal to the semilinear group ΓL(1, q ), thus proving that ΓL(1, q ) is 2-closed in the sense of Wielandt.

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Philipp Perepelitsky
November 6, 2008

### Abstract

If G is a group and H is a subgroup of G we write for the lattice of over-groups of H in G . It is an open question whether or not every finite lattice is isomorphic to some finite group interval lattice , where G is finite. We prove that no D Δ( m 1 , … , m t )-lattice and few M -lattices have the form , where G is an alternating or symmetric group of prime degree.

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Gerald Williams
November 7, 2008

### Abstract

The Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups are defined by the presentations G n ( m , k ) = 〈 x 1 , … , x n | x i x i+m = x i+k (1 ⩽ i ⩽ n )〉. These cyclically presented groups generalize Conway's Fibonacci groups and the Sieradski groups. Building on a theorem of Bardakov and Vesnin we classify the aspherical presentations G n ( m, k ). We determine when G n ( m, k ) has infinite abelianization and provide sufficient conditions for G n ( m, k ) to be perfect. We conjecture that these are also necessary conditions. Combined with our asphericity theorem, a proof of this conjecture would imply a classification of the finite Cavicchioli–Hegenbarth–Repovš groups.

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Martin R. Bridson
November 6, 2008

### Abstract

We consider finitely presented, residually finite groups G and finitely generated normal subgroups A such that the inclusion A ↪ G induces an isomorphism from the profinite completion of A to a direct factor of the profinite completion of G . We explain why A need not be a direct factor of a subgroup of finite index in G ; indeed G need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether A is a direct factor of a subgroup of finite index in G .

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Martin Hertweck, Eric Jespers
November 7, 2008

### Abstract

For a group G , let 𝓤 be the group of units of the integral group ring ℤ G . The group G is said to have the normalizer property if N 𝓤 ( G ) = Z(𝓤) G . It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which class-preserving automorphisms are inner automorphisms, Out c ( G ) = 1, have the normalizer property. Recently, Herman and Li have shown that Out c ( G ) = 1 for a finite Blackburn group G . We show that Out c ( G ) = 1 for the members G of certain classes of metabelian groups, from which the Herman–Li result follows. Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G , the group Z ∞ (𝓤) of hypercentral units of 𝓤 is contained in Z(𝓤) G .