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Karl W. Gruenberg, Peter A. Linnell
September 19, 2008

### Abstract

Let G be a group of the form , the free product of n subgroups, and let M be a ℤG -module of the form . We shall give formulae in various situations for d ℤG ( M ), the minimum number of elements required to generate M . In particular if C 1 , C 2 are non-trivial finite cyclic groups of coprime orders, and F/R ≅ G is the free presentation obtained from the natural free presentations of the two factors, then the number of generators of the relation module, d ℤG ( R/R ′), is 3. It seems plausible that the minimum number of relators of G should be 4, and this would give a finitely presented group with positive relation gap. However we cannot prove this last statement.

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J. R. J. Groves, D. H. Kochloukova, C. F. Rodrigues
September 19, 2008

### Abstract

We show that abelian-by-polycyclic groups of homological type FP 3 are virtually nilpotent-by-abelian; this generalizes earlier results of C.J.B. Brookes. We apply this to show that if G is the split extension of A by Q with A abelian, Q polycyclic and G of homological type FP 3 , then every quotient of G also has homological type FP 3 .

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V. V. Bludov, A. M. W. Glass
September 19, 2008

### Abstract

We prove that if G j are right orderable groups with non-trivial cyclic subgroups C j ( j = 1,2), then the (group) free product of G 1 and G 2 with C 1 and C 2 amalgamated is right orderable. We will deduce this from our main theorem that every lattice-ordered group can be embedded in one in which there are precisely four conjugacy classes. As a consequence of our method, we also show that every right ordered group can be embedded in a right ordered group in which any two non-identity elements are conjugate.

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M. J. Chasco, E. Martín-Peinador
September 19, 2008

### Abstract

We prove that every dense subgroup of a topological abelian group has the same ‘convergence dual’ as the whole group. By the ‘convergence dual’ we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence structure on the character group of a precompact group is discrete and therefore a non-compact precompact group is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least in the category of metrizable abelian groups it does.

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Tom De Medts, Katrin Tent
September 19, 2008

### Abstract

Moufang sets are split doubly transitive permutation groups, or equivalently, groups with a split BN-pair of rank one. In this paper, we study so-called special Moufang sets with abelian root groups, under the model-theoretic restriction that the groups have finite Morley rank. These groups have a natural base field, and we classify them under the additional assumption that the base field is infinite. The result is that the group is isomorphic to PSL 2 ( K ) over some algebraically closed field K .

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Hung Ngoc Nguyen, Pham Huu Tiep, Frank Himstedt
September 19, 2008

### Abstract

We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups 3 D 4 ( q ) in characteristic coprime to q to any proper subgroup is reducible.

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John N. Bray, Robert A. Wilson
September 19, 2008

### Abstract

It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, until now no explicit examples have been found. We present two explicit examples where the dimension is exactly 3. It remains an open question as to whether the dimension can be bigger than 3.

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David P. Cargo, Warwick de Launey, Martin W. Liebeck, Richard M. Stafford
September 19, 2008

### Abstract

In this paper, we consider finite groups G satisfying identities of the form . We focus on identities with r small, , and all coprime to the order of G . We show that for r = 2,3 and 5, G must be nilpotent. We also classify for r = 4,6 and 7, the special identities which can hold in non-nilpotent groups. Finally, we show that for r < 30, the group G must be solvable.

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Michael Geline
September 19, 2008

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Jinshan Zhang, Wujie Shi
September 19, 2008

### Abstract

For a finite group G and a non-linear irreducible complex character χ of G write υ (χ) = { g ∈ G |χ( g ) = 0}. There are two dual questions as follows: what are the finite groups G such that υ (χ) is a conjugacy class for all but one of the non-linear irreducible characters χ of G , and what are the finite groups G whose character table has the property that all but one of the columns have at most one zero entry. We completely answer the two questions using the classification of finite simple groups.

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Carlos A. M. André, Alejandro P. Nicolás
September 19, 2008

### Abstract

We study irreducible characters, and define and study supercharacters of the adjoint group of a finite radical ring. We extend work of Halasi on irreducible characters, and recent work of Diaconis and Isaacs on supercharacters of algebra groups defined over finite fields, and derive some consequences for the representation theory of compact p -adic algebra groups. We also define the notion of a fully ramified supercharacter, and consider orthogonal decompositions of a given supercharacter. Finally, we obtain a necessary and sufficient condition for a supercharacter to have a unique irreducible constituent, and illustrate the result in the case of the unitriangular group over a finite field.