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Paul Gartside, Michael Smith
November 20, 2009

### Abstract

The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds can be placed on its complexity as expressed by its scattered height.

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Cristóbal Rivas
November 20, 2009

### Abstract

We classify C -orderable groups admitting only finitely many C -orderings. We show that if a C -orderable group has infinitely many C -orderings, then it has uncountably many C -orderings, and none of these is isolated in the space of C -orderings. We carefully study the case of the Baumslag–Solitar group B (1, 2) and show that it has four C -orderings, each of which is bi-invariant, but that its space of left-orderings is homeomorphic to the Cantor set.

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V. Roman'kov
November 20, 2009

### Abstract

An algorithm is constructed that, when given an explicit presentation of a polycyclic group G , decides for any endomorphism ψ ∈ End( G ) and any pair of elements u , v ∈ G , whether or not the equation ( xψ ) u = vx has a solution x ∈ G . Thus it is shown that the problem of the title is decidable. Also we present an algorithm that produces a finite set of generators of the subgroup Fix ψ ( G ) ⩽ G of all ψ -invariant elements of G .

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Pedro Silva, Pascal Weil
November 20, 2009

### Abstract

We study the lattice of finite-index extensions of a given finitely generated subgroup H of a free group F . This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H . This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exponential and super-polynomial upper bound for the number of finite-index extensions of H , and we give a language-theoretic characterization of the lattice of finite-index subgroups of H . Finally, we give a polynomial-time algorithm to compute the malnormal closure of H .

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Raul Moreira Behs, Rudolf Maier
November 20, 2009

### Abstract

We show that the nilpotency of the metacyclic subgroups of a hypercyclic group implies the hypercentrality of the whole group. We also construct a center-free metabelian group all of whose polycyclic subgroups are abelian.

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Simion Breaz
November 20, 2009

### Abstract

For a self-small abelian group A of torsion-free rank 1, we characterize A -reflexive abelian groups which are induced by the pair of right adjoint contravariant functors Hom(–, A ) : Ab → Ab : Hom(–, A ).

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A. R. Jamali, M. Viseh
November 20, 2009

### Abstract

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G . We prove that if | G : A | ⩽ exp( A ), the exponent of A , then A contains a non-trivial normal subgroup of G . This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by |G : A| < Exp( G ), where Exp( G ) denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G . We will use these results to derive some facts about transitive permutation groups.

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Primož Moravec
November 20, 2009

### Abstract

We introduce the notion of a powerful action of a p -group upon another p -group. This represents a generalization of powerful p -groups introduced by Lubotzky and Mann in 1987. We derive some properties of powerful actions and study faithful powerful actions. We also prove that the non-abelian tensor product of powerful p -groups acting powerfully and compatibly upon each other is again a powerful p -group.

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Shripad M. Garge, Joseph Oesterlé
November 20, 2009

### Abstract

A Gelfand model for a finite group G is a complex linear representation of G that contains each of its irreducible representations with multiplicity one. For a finite group G with a faithful representation V , one constructs a representation which we call the polynomial model for G associated to V . Araujo and others have proved that the polynomial models for certain irreducible Weyl groups associated to their canonical representations are Gelfand models. In this paper, we give an easier and uniform treatment for the study of the polynomial model for a general finite Coxeter group associated to its canonical representation. Our final result is that such a polynomial model for a finite Coxeter group G is a Gelfand model if and only if G has no direct factor of the type W ( D 2 n ), W ( E 7 ) or W ( E 8 ).

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Matthew Dyer
November 20, 2009

### Abstract

For a finitely generated subgroup W ′ of a Coxeter system ( W , S ), there are finitely generated reflection subgroups W 1 , . . . , W n of W , each containing W ′, such that any reflection subgroup of W containing W ′ contains one of the W i as a standard parabolic subgroup. The canonical Coxeter generators of the W i , and an expression for the parabolic closure of W ′ as a W -conjugate of a standard parabolic subgroup of W , may be effectively determined.

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Xinjian Zhang, Xianhua Li
August 31, 2009

### Abstract

Let M be a maximal subgroup of a finite group G . The order of a chief factor H/K such that H is a minimal supplement to M in G is called the normal index of M , and ( M ∩ H )/ K is called a c -section of M . Using the concepts of normal index and c -section, we obtain characterizations for finite groups of the solvable, p -supersolvable and supersolvable groups.