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Dessislava H. Kochloukova
July 14, 2009

### Abstract

We show that limit groups are free-by-(torsion-free nilpotent) and have non-positive Euler characteristic. We prove that for any non-abelian limit group the Bieri–Neumann–Strebel–Renz Σ-invariants are the empty set. Let s ⩾ 3 be a natural number and G be a subdirect product of non-abelian limit groups intersecting each factor non-trivially. We show that the homology groups of any subgroup of finite index in G , in dimension i ⩽ s and with coefficients in ℚ, are finite-dimensional if and only if the projection of G to the direct product of any s of the limit groups has finite index. The case s = 2 is a deep result of M. Bridson, J. Howie, C. F. Miller III and H. Short.

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S. V. Ivanov
April 17, 2009

### Abstract

We introduce and study a property of groups which could be regarded as a broad generalization of the Cauchy–Davenport theorem and which turns out to be useful in investigation of the intersection of subgroups in free products of groups.

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Paul Gartside, Michael Smith
July 14, 2009

### Abstract

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.

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B. A. F. Wehrfritz
July 14, 2009

### Abstract

Suppose that N is a normal subgroup of the Chernikov group G such that the direct product of N and G / N is isomorphic to G . We prove that N is a direct factor of G . This extends and uses work of Ayoub on finite groups and supplements work of Nikolov and Segal on polycyclic-by-finite groups.

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Josh Wiscons
July 14, 2009

### Abstract

In this paper we study special Moufang sets 𝕄( U , τ ), with U abelian, under the additional restriction that they have finite Morley rank. Our result states that the little projective group of such a Moufang set must be isomorphic to PSL 2 ( K ) for an algebraically closed field K provided that U has characteristic 2 and that infinitely many endomorphisms of U centralize the Hua subgroup. This complements a result of De Medts and Tent that addresses the odd characteristic case.

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Russell D. Blyth, Francesco Fumagalli, Marta Morigi
July 14, 2009

### Abstract

We study the non-abelian tensor square G ⊗ G for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G / G ′ so that G ⊗ G is isomorphic to the direct product of ∇( G ) and the non-abelian exterior square G ∧ G . For any group G , we characterize the non-abelian exterior square G ∧ G in terms of a presentation of G . Finally, we apply our results to some classes of groups, such as the classes of free solvable and free nilpotent groups of finite rank, and some classes of finite p -groups.

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Matthew J. Dyer
July 22, 2009

### Abstract

Admissibility of a subset S ′ of a Coxeter system ( W, S ) is a condition implying that S ′ is the set of Coxeter generators of a Coxeter subgroup W ′ of W , in such a way that the root system of W ′, as permutation set, abstractly embeds in that of W . We give an algorithm determining whether a subset S ′ is admissible, in terms of a (previously known) finite state automaton which is constructed using the set of elementary roots of Brink and Howlett.

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Tom Edgar
July 22, 2009

### Abstract

Any normal reflection subgroup of a Coxeter system ( W, S ) is a factor in a semidirect product decomposition of W as described by Bonnafé and Dyer. Namely, S is the union of two subsets I and J such that no element of I is conjugate to an element of J , is the subgroup generated by W I -conjugates of elements of J , and W is the semidirect product of W I by . This note describes the reduced expressions of elements of the form wxw –1 with w ∈ W I and x ∈ W J in terms of reduced expressions of x and a suitable element of W I .

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David I. Stewart
July 14, 2009

### Abstract

Let G ≔ G 2 ( K ) be a simple algebraic group of type G 2 defined over an algebraically closed field K of characteristic p > 0. Let σ denote a standard Frobenius automorphism of G such that G σ ≅ G 2 ( q ) with q ⩾ 4. In this paper we find all reductive subgroups of G and quasi-simple subgroups of G σ in the defining characteristic. Our results extend the complete reducibility results of [Liebeck and Seitz, Mem. Amer. Math. Soc. 121: 580, 1996, Theorem 1].

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Michael A. Jackson
July 14, 2009

### Abstract

The generalized symmetric groups are defined to be the groups G ( n, m ) = ℤ m ≀ Σ m where n, m ∈ ℤ + . The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation-preserving automorphisms. The present paper extends work on the strong symmetric genus by Conder, who studied the symmetric groups, which are the groups G ( n , 1), and the author, who studied the hyperoctahedral groups, which are the groups G ( n , 2). We determine the strong symmetric genus of the groups G ( n , 3).

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Pavel Shumyatsky
July 14, 2009

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A. Ballester-Bolinches, R. Esteban-Romero, Yangming Li
July 14, 2009

### Abstract

The aim of this paper is to characterize the classes of groups in which every subnormal subgroup is normal, permutable, or S -permutable in terms of the embedding of subgroups (or subgroups of prime-power order) in their normal, permutable, or S -permutable closure.

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Clara Franchi, Mario Mainardis, Ronald Solomon
July 14, 2009