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John R. Britnell, Mark Wildon
June 15, 2009

### Abstract

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c ∈ C and d ∈ D such that cd = dc . In the case where G has a normal subgroup H such that G / H is cyclic, two theorems are proved concerning the distribution, between cosets of H , of pairs of conjugacy classes of G related by ~. One of the proofs involves an application of the famous marriage theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.

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Jean-Baptiste Gramain
June 15, 2009

### Abstract

Following the work of B. Külshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the -defect of characters of the symmetric group 𝔖 n , where is an arbitrary integer. We prove that the -defect is given by an analogue of the hook-length formula, and use it to prove, when , an -version of the McKay conjecture in 𝔖 n .

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James B. Wilson
June 15, 2009

### Abstract

A Las Vegas polynomial-time algorithm is given to find a central decomposition of maximum size for a finite p -group of class 2. The proof introduces an associative *-ring as a tool for studying central products of p -groups. This technique leads to a translation of the problem into classical linear algebra which can be solved by application of the MeatAxe and other established module-theoretic algorithms. When p is small, our algorithm runs in deterministic polynomial time.

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Jizhu Nan, Jing Zhao
June 15, 2009

### Abstract

In this paper, we define B to be a certain subgroup in a classical group over a finite field and use it to construct non-classical BN-pairs. We also show that this construction is different from the classical one. As an application, we give transcendence bases for the fields of rational invariants of their subgroups B and N .

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Peter J. Witbooi
June 15, 2009

### Abstract

For a relatively prime pair of natural numbers n, u , let G ( n ; u ) = 〈 a , b | a n = 1, bab –1 = a u 〉. For groups H of the type G ( n ; u ) × G ( m ; u ) we determine the set χ( H ) of all isomorphism classes of groups K with the property that K × ℤ ≅ H × ℤ. The set χ( H ) carries a certain group structure which coincides with the Hilton–Mislin genus group structure when H is nilpotent. We compute the group χ( H ).

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Stewart Stonehewer, Giovanni Zacher
June 15, 2009

### Abstract

Given a group G and subgroups X ⩾ Y , with Y of finite index in X , then in general it is not possible to determine the index | X : Y | simply from the lattice of subgroups of G . For example, this is the case when G has prime order. The purpose of this work is twofold. First we show that in any group, if the indices | X : Y | are determined for all cyclic subgroups X , then they are determined for all subgroups X . Second we show that if G is a group with an ascending normal series with factors locally finite or abelian, and if the Hirsch length of G is at least 3, then all indices | X : Y | are determined.

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Martin Hertweck, Christian R. Höfert, Wolfgang Kimmerle
June 15, 2009

### Abstract

Let G denote the projective special linear group PSL(2, q ), for a prime power q . It is shown that a finite 2-subgroup of the group V(ℤ G ) of augmentation 1 units in the integral group ring ℤ G of G is isomorphic to a subgroup of G . Furthermore, it is shown that a composition factor of a finite subgroup of V(ℤ G ) is isomorphic to a subgroup of G .

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Agnese Ilaria Telloni
June 15, 2009

### Abstract

We introduce a family of cyclically presented groups defined by five parameters, suggested by fundamental groups of certain closed 3-manifolds. This family includes groups studied by Sidki in [On the fundamental groups of 3-manifolds of Lins–Mandel]. We determine several algebraic properties of our family of groups, and give a geometrical interpretation of such groups for particular choices of parameters.

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Siddhartha Gadgil, Geetanjali Kachari
June 15, 2009

### Abstract

We give a description, modulo torsion, of the cup product on the first cohomology group in terms of the descriptions of the second homology group due to Hopf and Miller.

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Simon Thomas
June 15, 2009

### Abstract

There does not exist a Borel way of selecting an isomorphism class within each commensurability class of finitely generated groups.

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Vincenzo Marra
June 15, 2009

### Abstract

We characterize the Lebesgue state of a free finitely generated unital lattice-ordered abelian group G in terms of its value at each element of each basis of G . This significantly strengthens one of the main results of our previous paper (co-authored by D. Mundici) with the same title as the present one. As a consequence of independent interest, we obtain a state-theoretic characterization of free finitely generated objects in the category of unital lattice-ordered abelian groups and their unit-preserving lattice-group homomorphisms.

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Anton A. Klyachko
June 15, 2009

### Abstract

Suppose that G is a non-trivial torsion-free group and w is a word in the alphabet such that the word w ′ ∈ F ( x 1 , . . . , x n ) obtained from w by erasing all letters belonging to G is not a proper power in the free group F ( x 1 , . . . , x n ). We show how to reduce the study of the relative presentation Ĝ = 〈 G , x 1 , x 2 , . . . , x n | w = 1〉 to the case n = 1. It turns out that any such ‘ n -variable’ group Ĝ can be obtained from similar ‘one-variable’ groups by using an explicit construction similar to the wreath product. As an illustration, we prove that, for n ⩾ 2, the centre of Ĝ is always trivial. For n = 1, the centre of Ĝ is also almost always trivial; there are several exceptions, and all of them are known.

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Le Thi Giang
June 22, 2009

### Abstract

We prove that if G is a torsion-free group and w is a word in the alphabet G ⊔ { t ±1 } with exponent sum 1 in t , then the group 〈 G, t | w k = 1〉, for k ⩾ 2, is relatively hyperbolic with respect to G .

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A. Ballester-Bolinches, R. Esteban-Romero, M. Ragland
June 15, 2009