Universal commutator relations : doctoral thesis
ID Jezernik, Urban (Avtor), ID Moravec, Primož (Mentor) Več o mentorju...

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Izvleček
We 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.

Jezik: Angleški jezik fake degree conjecture, commutator relation, Bogomolov multiplier, commutativity preserving extension, commuting probability Doktorsko delo/naloga 2.08 - Doktorska disertacija FMF - Fakulteta za matematiko in fiziko Ljubljana [U. Jezernik] 2016 134 str. 20.500.12556/RUL-97405 512(043.3) 17702233 24.10.2017 1723 436 Kopiraj citat

## Sekundarni jezik

Jezik: Slovenski jezik Univerzalne komutatorske relacije Raziskujemo relacije med komutatorji v abstraktnih grupah. Med njimi izpostavimo univerzalne relacije, ki so zgolj posledice algebraičnih manipulacij. Osredotočimo se na komutatorske relacije, ki niso take. Te pristno netrivialne neuniverzalne komutatorske relacije je mogoče zbrati v abelovo grupo, imenovano multiplikator Bogomolova. Ta objekt ima tukaj osrednjo vlogo. Še posebej nas zanima vprašanje njegove trivialnosti in posedovanje nekakšnega nadzora nad njegovim obnašanjem. V disertaciji predstavimo razne vidike tega. Po krajšem pregledu motivacije za študij multiplikatorja Bogomolova raziščemo nekaj njegovih osnovnih lastnosti. Podamo mnogo primerov grup s trivialnimi in netrivialnimi multiplikatorji Bogomolova ter prikažemo različne metode. Predstavimo kohomološko interpretacijo multiplikatorja Bogomolova, s čimer vzpostavimo povezavo med komutatorskimi relacijami in razširitvami grup, ki ohranjajo komutativnost. Multiplikator Bogomolova je univerzalen objekt, ki parametrizira takšne razširitve dane grupe. Razvijemo teorijo krovnih grup. Te konstrukcije uporabimo za izgradnjo učinkovitega algoritma za računanje multiplikatorjev Bogomolova končnih rešljivih grup. Nadalje raziščemo grupe, ki so minimalne glede na posedovanje neuniverzalnih komutatorskih relacij. Pridobljene rezultate uporabimo za študij problema trivialnosti multiplikatorja Bogomolova iz verjetnostnega vidika. Podamo eksplicitno spodnjo mejo za verjetnost komutiranja, ki zagotovi trivialnost multiplikatorja Bogomolova. Izpeljemo relativne strukturne meje v zvezi z multiplikatorjem Bogomolova. Verjetnost komutiranja povežemo z razširitvami, ki ohranjajo komutativnost, s čimer omejimo multiplikator Bogomolova v odvisnosti od verjetnosti komutiranja dane grupe. Nazadnje izkoristimo še eno znano pojavitev multiplikatorja Bogomolova, da podamo negativen odgovor na Isaacsovo domnevo o stopnjah karakterjev končnih grup, ki izhajajo iz nilpotentnih asociativnih algeber. Domnevi se približamo z grupami, ki izhajajo iz modularnih grupnih kolobarjev. Za opaženo neregularnost ponudimo tudi bolj konceptualno razlago z vidika algebraičnih grup. Pokažemo, da lahko elemente multiplikatorja Bogomolova vidimo kot racionalne točke na neki komutatorski raznoterosti. domneva o lažnih stopnjah, komutatorska relacija, multiplikator Bogomolova, razširitev, ki ohranja komutativnost, verjetnost komutiranja

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