Volume 14, Issue 6

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.

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.

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 .

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.

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].

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 à 2 .

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.

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.

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.

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.