Volume 21, Issue 6

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.

We study the structure of various groups generated by (left) multiplications in alternative division algebras, and construct epimorphisms onto orthogonal and special orthogonal groups. Throughout, we include the characteristic two case in our treatment.

We prove that an abstract oval is an abstract conic if and only if all of its involutions are regular and all of its secant lines are Pascalian. In doing so, we provide a new characterisation of the rank 1 projective general linear groups over arbitrary fields.

We show that the tower of profinite completions of a nonstrongly complete profinite group continues indefinitely.

Let G be a finite group, p a prime and c∈{0,1,…,p-1}{c\in\{0,1,\ldots,p-1\}}. Suppose that the degree of every nonlinear irreducible character of G is congruent to c modulo p . If here c=0{c=0}, then G has a normal p -complement by a well known theorem of Thompson. We prove that in the cases where c≠0{c\neq 0} the group G is solvable with a normal abelian Sylow p -subgroup. If p≠3{p\neq 3} then this is true provided these character degrees are congruent to c or to -c{-c} modulo p .

Let G be a locally finite group and let F⁢(G){F(G)} be the Hirsch–Plotkin radical of G . Let S denote the full inverse image of the generalized Fitting subgroup of G/F⁢(G){G/F(G)} in G . Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k . The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient G/S{G/S} contains an abelian subgroup of finite index bounded in terms of k . We disprove this statement and prove a weak analogue of it.

In this note we give two characterizations of finite nilpotent groups. First, we show that a finite group G is not p -nilpotent if and only if it contains two elements of order qk{q^{k}}, for q a prime different than p , whose product has order p or possibly 4 if p=2{p=2}. We also show that the set of words on two variables where the total degree of each variable is ±1{\pm 1} can be used to characterize finite nilpotent groups. Using this characterization we show that if a finite group is not nilpotent, then there is a word map of specified form for which the corresponding probability distribution is not uniform.