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Arjeh M. Cohen, Hans Cuypers, Ralf Gramlich
December 18, 2008

### Abstract

In this article we consider a sub-geometry of the D 4 building geometry whose flags of type {1, 3, 4} are exactly those which are opposite to their image under a triality on D 4 , while the lines of are certain so-called skew lines (see Definition 3.4). We prove that this rank four geometry admits the group G 2 as a flag-transitive group of automorphisms. Moreover, if the underlying field contains at least three elements, the geometry is simply connected. Accordingly, we obtain an amalgam presentation of G 2 via the rank one and two parabolics of the action of G 2 on .

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Rex Dark, Arnold D. Feldman, María Dolores Pérez-Ramos
December 18, 2008

### Abstract

In response to a question of Doerk and Hawkes [Finite soluble groups, de Gruyter, 1992, p. 553], we shall obtain characterizations of the injectors of a finite solvable group (without recourse to the concept of a Fitting set), and we also answer in the negative a question in [Dark and Feldman, J. Group Theory 9: 2006, p. 785].

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Antonio Beltrán, María José Felipe
December 18, 2008

### Abstract

Let G be a finite group and suppose that the set of conjugacy class sizes of G is {1, m , mn }, where m, n > 1 are coprime. We prove that m = p for some prime p dividing n – 1. We also show that G has an abelian normal p -complement and that if P is a Sylow p -subgroup of G , then | P′ | = p and | P / Z ( G ) p | = p 2 . We obtain other properties and determine completely the structure of G .

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Karl Heinrich Hofmann, Karl-Hermann Neeb
December 18, 2008

### Abstract

It is shown that a closed almost solvable subgroup of an almost connected locally compact group is compactly generated. In particular, every discrete solvable subgroup of a connected locally compact group is finitely generated.

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Simon Thomas
December 18, 2008

### Abstract

There does not exist an isomorphism-invariant Borel version of the Higman–Neumann–Neumann embedding theorem.

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Martin Kassabov, Nikolay Nikolov
December 18, 2008

### Abstract

We give a new and self-contained proof of a theorem of Linnell and Warhurst that d ( G ) – d ( Ĝ ) ⩽ 1 for finitely generated virtually torsion-free soluble minimax groups G . We also give a simple sufficient condition for the equality d ( G ) = d ( Ĝ ) to hold when G is virtually abelian.

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V. V. Bludov
December 18, 2008

### Abstract

Two examples of finitely presented soluble groups are constructed. The first is a metabelian residually torsion-free-nilpotent group G ≅ F / R having free central extension F /[ F, R ] which is not residually nilpotent and the second is a centrally ordered group that does not belong to the variety of lattice-ordered groups generated by all nilpotent lattice-ordered groups.

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Giovanni Cutolo, Howard Smith, James Wiegold
October 20, 2008

### Abstract

We characterize groups in which every abelian subgroup has finite index in its characteristic closure. In a group with this property every subgroup H has finite index in its characteristic closure and there is an upper bound for this index over all subgroups H of G . For every prime p we construct groups G with this property that are infinite nilpotent p -groups of class 2 and exponent p 2 in which G′ = Z ( G ) is finite and Aut G acts trivially on G / G′ . We also characterize abelian groups with the dual property that every subgroup has finite index over its characteristic core.

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Nicholas W. M. Touikan
January 19, 2009

### Abstract

Using the theory developed by Kharlampovich and Miasnikov, and independently by Sela, to describe the set of homomorphisms from a finitely generated group G into a free group F , we describe the solutions to equations of the form w ( x, y ) = u with coefficients from F and unknowns x, y , where u lies in F and w ( x, y ) is a word in { x, y } ±1 . We also give an example of a single equation whose solutions cannot be described with only one ‘level’ of automorphisms.