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A. M. W Glass
February 22, 2008

### Abstract

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem. A finitely generated lattice-ordered group has soluble word problem if and only if it can be ℓ-embedded in an ℓ-simple lattice-ordered group that can be ℓ-embedded in a finitely presented lattice-ordered group. The proof uses permutation groups, a technique of Holland and McCleary, and the ideas used to prove the lattice-ordered group analogue of Higman's embedding theorem.

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Joe L Mott, Muneer A Rashid, Muhammad Zafrullah
February 22, 2008

### Abstract

Throughout let G = ( G ,+,≤, 0) denote a Riesz group, where + is not necessarily a commutative operation. Call x ∈ G homogeneous if x > 0 and for all h, k ∈ (0, x ] there is t ∈ (0, x ] such that t ≤ h, k . In this paper we develop a theory of factoriality in Riesz groups based on the fact that if x ≤ G and x is a finite sum of homogeneous elements then x is uniquely expressible as a sum of finitely many mutually disjoint homogeneous elements. We then compare our work with existing results in lattice-ordered groups and in (commutative) integral domains.

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John Murray
February 22, 2008

### Abstract

Each p -block of a finite group has an associated defect group, which is a p -subgroup of the group. Each real 2-block has, in addition, an associated extended defect group , which is a 2-subgroup of the group that contains a defect group as a subgroup of index at most 2. We consider the possible extended defect groups of a real 2-block that has a cyclic or a Klein-four defect group. In each case we describe the modules in the block that are components in the permutation module of the group acting by conjugation on its involutions. We also determine the Frobenius–Schur indicators of the irreducible characters in the block.

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Victor Bovdi, Martin Hertweck
February 22, 2008

### Abstract

We confirm a conjecture of Zassenhaus about rational conjugacy of torsion units in integral group rings for a covering group of the symmetric group S 5 and for the general linear group GL(2, 5). The first result, together with others from the literature, settles the conjugacy question for units of prime-power order in the integral group ring of a finite Frobenius group.

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Luis Ribes
February 22, 2008

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Ahmet Arıkan
February 22, 2008

### Abstract

Let G be a minimal non-solvable Fitting p -group. Our main results are that G has uncountbly many subgroups and that G has a homomorphic image with a proper subgroup such that () = 1. We also define the solvabilizer of a solvable group and consider the minimal non-solvability problem in another context.

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Marcel Herzog, Patrizia Longobardi, Mercede Maj
February 22, 2008

### Abstract

Groups with a finite number of conjugacy classes of infinite size are investigated in this paper and are called -groups. The structure of groups in this class can be very complicated. Non- FC periodic locally graded -groups are characterized in Theorem 10. Groups with a single infinite conjugacy class are characterized in Theorem 3. The final sample result of this paper is from Theorem 9: if G is a -group and F = FC( G ), then [ G : FC G ( F )] < ∞.

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Olga Kharlampovich, Alexei Myasnikov, Vladimir Remeslennikov, Denis Serbin
February 22, 2008

### Abstract

In this paper we study exponential extensions of G which are finitely generated G -subgroups of the ℤ[ t ]-completion G ℤ[ t ] of a given CSA-group G . Using Bass–Serre theory we prove that exponential extensions of G can be obtained from subgroups of G by free constructions of a special type. As an application of this technique we describe the cohomological and homological dimensions of finitely generated fully residually free groups and give an algorithm to compute them.

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Leo P Comerford, Charles C Edmunds
February 22, 2008

### Abstract

We consider equations of the form W ( x, y ) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G , one can effectively determine whether or not the equations have solutions in G . We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W ( x, y ) = U has only finitely many solutions, up to the action of the stabilizer of W ( x, y ) in Aut(〈 x, y ; 〉).