Banca de DEFESA: João Pedro Papalardo Azevedo

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : João Pedro Papalardo Azevedo
DATE: 03/02/2023
TIME: 14:00
LOCAL: Departamento de Matemática
TITLE:

Probabilidade de comutação em grupos compactos

KEY WORDS:

Commuting probability, Haar measure, compact groups, monothetic
subgroups.

PAGES: 82
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

Let G be a compact topological group with a closed subgroup K and normalized
Haar measures and , respectively. Consider the closed subset C = {(x,y) K G | xy=yx} of
K G and define the relative commuting probability of K in G by Pr(K,G) = ()(C). This
value represents the probability of choosing at random an element of K and one of G
that commute. If K = G, we get the commuting probability of G, a measure of how
close to be abelian the group is. For years, the influence of Pr(G) and Pr(K,G) on the
structure of G has been studied. For example, a theorem of P.M. Neumann [40]
ensures that, if G is finite and is a positive number, Pr(G) implies that G has a
subgroup H such that [G:H] and |H'| are -bounded. Our goal is to study similar
properties concerning relative commuting probability.

In [9], Detomi and Shumyatsky prove structural resuts about a finite group G
having a subgroup K such that Pr(K,G) . They prove that there exist sungroups T of G
and B of K such that the indices [G:T] and [K:B] and the order of [T,B] are -bounded.
We extend this result to compact groups and prove corollaries of it.
If G is a topological group and x G, denote by <x> the closed subgroup
generated by x. We prove that, if Pr(<x>, G) for every x in a closed subgroup K of G,
then there are an open subgroup T of G and an integer such that the index [G:T] and
the number are -bounded and . This result represents a probabilistic interpretation of
the notion of exponent in a group. Several corollaries are proved, all related to the
notion of exponent. Finally, we consider the more general situation where Pr(<x>, G) is
positive for all x in K G. We prove that G has an open subgroup T in such a way that
every x K has a power , where l is not necessarily fixed, centralizing T.

BANKING MEMBERS:
Presidente - 1224071 - PAVEL SHUMYATSKY
Interna - 1994292 - CRISTINA ACCIARRI
Interno - 1209688 - RAIMUNDO DE ARAUJO BASTOS JUNIOR
Externa à Instituição - MARTA MORIGI - UNIBO-IT
Externa à Instituição - ELOISA DETOMI - PADOVA