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Anna Luisa Gilotti, Luigi Serena
February 10, 2011

### Abstract

In this paper we generalize and unify several results proved in recent papers about the existence of normal p -complements and conditions on supersolubility. Moreover a counterexample is given to a question in [Guo and Wei, J. Group Theory 13: 267–276, 2010] and it is proved that a finite group is 2-nilpotent if the cyclic subgroups of order less or equal than four are strongly closed.

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Andrea Lucchini
February 10, 2011

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Michael Sun
March 15, 2011

### Abstract

We give a semi-direct product decomposition of the point stabilisers for the enhanced and exotic nilpotent cones. In particular, we arrive at formulas for the number of points in each orbit over a finite field, which is in accordance with a recent conjecture of Achar and Henderson.

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Zoltán Halasi, Péter P. Pálfy
March 15, 2011

### Abstract

A famous open problem due to Graham Higman asks if the number of conjugacy classes in the group of n × n unipotent upper triangular matrices over the q -element field can be expressed as a polynomial function of q for every fixed n . We consider the generalization of the problem for pattern groups and prove that for some pattern groups of nilpotency class two the number of conjugacy classes is not a polynomial function of q .

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Gil Kaplan
February 10, 2011

### Abstract

The Wielandt subgroup of a group G is the intersection of the normalizers of all the subnormal subgroups of G . A T-group is a group in which all the subnormal subgroups are normal, or, equivalently, a group coinciding with its Wielandt subgroup. We investigate the Wielandt subgroup of finite solvable groups and, in particular, find new properties and characterizations (see Theorems 1, 2 and Corollaries 4, 6) for this subgroup in the case that G is metanilpotent. Furthermore, we provide new characterizations for finite solvable T -groups in Theorem 7.

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Kamal Aziziheris
February 10, 2011

### Abstract

We prove that if G is a solvable group with cd( G ) = {1, m, p, q, mp, mq }, where p and q are distinct primes and m > 1 is an integer not divisible by p or q , then G = A × B , where cd( A ) = {1, p, q } and cd( B ) = {1, m }. This generalizes [Lewis, J. Algebra 206: 235–260, 1998, Theorem A].

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Guyan Robertson
March 15, 2011

### Abstract

Let Γ be a torsion-free discrete group acting cocompactly on a two dimensional euclidean building Δ. The centralizer of an element of Γ is either a Bieberbach group or is described by a finite graph of finite cyclic groups. Explicit examples are computed, with Δ of type Ã 2 .

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Koji Nuida
March 15, 2011

### Abstract

Let W be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer Z W ( W I ) of an arbitrary parabolic subgroup W I into the center of W I , a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. By using our description of Z W ( W I ), we will be able to reveal a further strong property of the action of the third factor on the second factor, in particular on the finite irreducible components of the second factor.

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Karl H. Hofmann, Sidney A. Morris
March 15, 2011

### Abstract

In the book “The Lie Theory of Connected Pro-Lie Groups” the authors proved the local splitting theorem for connected pro-Lie groups. George A. A. Michael subsequently proved this theorem for almost connected pro-Lie groups. Here his result is proved more directly using the machinery of the aforementioned book.

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Carl Droms
November 6, 2011

### Abstract

An embedding of an infinite Cayley graph in the two–sphere has either one, two, or an infinite number of essential accumulation points of vertices. We obtain a list of group presentations which includes every group possessing a Cayley graph that can be embedded in the two–sphere with two essential accumulation points of vertices.

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Benjamin M. S. Martin
March 15, 2011

### Abstract

Let F be a finitely generated group and let G be a linear algebraic group over an algebraically closed field k . Let R( F, G ) be the variety of representations of F in G . Given a finite set of generators Δ for F , we define a compactification R Δ ( F , ) of R( F, G ). The compactification depends on the choice of generators. If F ′ is another finitely generated group with a finite set of generators Δ′ and ϕ : F ′ → F is a homomorphism, then there is an induced morphism of varieties ϕ # : R( F, G ) → R( F ′, G ). We prove that if ϕ (Δ′ ∪ {1}) ⊆ Δ ∪ {1}, then ϕ # extends to a morphism from R Δ ( F , ) to R Δ′ ( F ′, ). We study the morphisms arising in this way from a group extension 1 → N → F → Q → 1.