Volume 17, Issue 5

If ( W , S ) is a right-angled Coxeter system and W has no ℤ 3 subgroups, then it is shown that the absence of an elementary separation property in the presentation diagram for ( W , S ) implies all CAT(0) spaces acted on geometrically by W have locally connected CAT(0) boundary. It was previously known that if the presentation diagram of a general right-angled Coxeter system satisfied the separation property, then all CAT(0) spaces acted on geometrically by W have non-locally connected boundary. In particular, this gives a complete classification of the right-angled Coxeter groups with no 3-flats and with locally connected boundary.

We present an algorithm for solving the conjugacy search problem in the four-strand braid group. The computational complexity is cubic with respect to the braid length.

We define the notion of a commutator socle-regular Abelian p -group. After establishing some crucial properties of commutator socle-regularity, we investigate its relationship with socle-regularity, strong socle-regularity and projection socle-regularity.

The minimal faithful permutation degree of a finite group G , denoted by μ( G ), is the least non-negative integer n such that G embeds inside Sym( n ). In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the group.

Call a compact, connected, simple Lie group G adjoint simple if it has trivial center. Let C ⊂ G be a nontrivial conjugacy class, e ∈ G the identity element of G . We prove the existence of a bound N ∈ ℕ, depending on G but not C , such that e lies in the interior of C n for all n ≥ N . We then prove that a disk D ⊂ ℂ of radius less than 1, properly contained in the unit disk D 1 and tangent to D 1 at z = 1, contains the image of every normalized character χ(e)-1χ${\chi (e)^{-1}\chi }$ of G .

A set of subgroups of a group is said to be a partition if every nonidentity element belongs to one and only one subgroup in this set. The study of groups with partition dates back to a paper by Miller published in 1906 ([Bull. Amer. Math. Soc. 17 (1906/1907), 446–449]). In this paper we study the structure of loops that are partitioned by subgroups.

We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F,h]=F${[ F,h] =F}$ for all nonidentity elements h ∈ H . We prove that any irreducible nontrivial F H -module for a Frobenius-like group F H of odd order over an algebraically-closed field has an H -regular direct summand if either F is fixed-point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F . Some consequences of this result are also derived.

Erratum to http://dx.doi.org/10.1515/jgt-2014-0002.

It is known that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.

In this paper we consider finite groups G such that every non-cyclic maximal subgroup in its Sylow subgroups is subnormal in G . In particular, we prove that such solvable groups have an ordered Sylow tower.

We will determine all infinite 2-locally finite groups and all infinite 2-groups with planar subgroup graph, and show that infinite groups that satisfy the chain conditions and contain an involution do not have planar embeddings. We also determine all connected outer-planar groups and outer-planar groups satisfying the chain conditions. As a result, we obtain all planar groups that are direct products of connected groups.

In this paper we are concerned with finite soluble groups G admitting a factorisation G=AB${G=AB}$, with A and B proper subgroups having coprime order. We are interested in bounding the Fitting height of G in terms of some group-invariants of A and B , including the Fitting heights and the derived lengths.