Universal commutator relationsJezernik, Urban (Avtor)
Moravec, Primož (Mentor)
fake degree conjecturecommutator relationBogomolov multipliercommutativity preserving extensioncommuting probabilityWe study relations between commutators in abstract groups. Of these, we expose universal relations that are consequences of purely algebraic manipulations. We focus on the commutator relations that are not of this sort. It is possible to collect the truly nontrivial nonuniversal commutator relations into an abelian group called the Bogomolov multiplier, which is the fundamental object of interest here. It is of particular interest to determine whether or not this object is trivial, or at least to have some control over it. The present thesis is an exposition of various aspects of this. After presenting a brief historical motivation for studying this object, we explore some of its basic properties. Many examples of groups with both trivial and nontrivial Bogomolov multipliers are given, illustrating different techniques. We present a cohomological interpretation of the Bogomolov multiplier, which makes it possible to relate commutator relations to the study of commutativity preserving extensions of groups. The Bogomolov multiplier is a universal object parameterizing such extensions. A theory of covering groups is developed. These constructions are then used to produce an effective algorithm for computing Bogomolov multipliers of finite solvable groups. We further inspect groups that are minimal with respect to possessing nonuniversal commutator relations. The results of this are used to study the problem of triviality of the Bogomolov multiplier from the probabilistic point of view. We give an explicit lower bound for commuting probability that ensures triviality of the Bogomolov multiplier. Relative structural bounds on the Bogomolov multiplier are presented. By relating commuting probability to commutativity preserving extensions, these bounds are used to bound the Bogomolov multiplier relative to the commuting probability. We end by making use of another known apparition of the Bogomolov multiplier to give a negative answer to a conjecture of Isaacs about character degrees of finite groups arising from nilpotent associative algebras by adjoining a unit. The conjecture is tackled by considering such groups that arise from modular group rings. A more conceptual explanation for the observed irregular behavior is provided by looking at the situation from the point of view of algebraic groups. We show how elements of the Bogomolov multiplier can be seen as rational points on a certain commutator variety.[U. Jezernik]20162017-10-24 14:38:44Doktorsko delo/naloga97405sl