Group Theory session:
Thursday, August 8
14.30-15.30
Csaba Schneider (Federal University of Minas Gerais, Brazil)
A new look at permutation groups of simple diagonal type
Permutation groups of simple diagonal type form one of the classes of (quasi)primitive permutation groups identified by the O'Nan-Scott Theorem. They also occur among the maximal subgroups of alternating and symmetric groups. Until now, they were not considered as a geometric class in the sense that they were not viewed as stabilizers of geometric or combinatorial objects. In this talk I will report on some new research, carried out in collaboration with Cheryl Praeger, Peter Cameron and Rosemary Bailey, whose results show that these groups can also be viewed as full stabilizers of certain combinatorial structures. I will also show that a permutation group of simple diagonal type is the automorphism group of a graph which is constructed as the edge union of Hamming graphs. The results hold also for infinite permutation groups.
15.40- 16.10
Theo Zapata (Universidade de Brasília, Brasil)
Profinite groups in which the centralizer of any non-identity element is abelian
Groups in the title (\textit{i.e.}, profinite CA-groups) are also known as profinite commutativity-transitive groups. In this talk I shall present a dichotomy theorem obtained with P.~Shumyatsky and P.~Zalesskii (2019, Israel J. Math, v.~230): Any profinite CA-group has a finite index closed subgroup that is either abelian or pro-$p$.
16.10-16-40
Sheila Campos Chagas (University of Brasilia, Brazil)
Subgroup Conjugacy Separability
We will discuss the Subgroup Conjugacy Separability property introduced by O. Bogopolski and F. Grunewald. A group $G$ is said to be subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups $H$ and $K$ of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. This is a joint work with Pavel Zalesskii.
17.10-18.10
Pavel Zalesskii (University of Brasilia, Brazil)
Virtually free pro-p groups and p-adic represantations
We shall discuss the connection between of a pro-$p$ group $G=F\rtimes K$, where $F$ is free of rank $n$ and $K$ is finite, and p-adic representations of $K$ in $GL_n(\mathbb{Z}_p)$.
18.20-18.50
Ismael Gutiérrez García (Universidad del Norte, Colombia)
The Strong Containment for Saturated Formations of Finite Soluble Groups
All objects called groups will be supposed to belong to the class $\mathfrak{S}$ of the finite soluble groups. Let $\mathfrak{F}$ and $\mathfrak{H}$ be saturated formations with $\mathfrak{F}\subseteq\mathfrak{H}$. The fact that $\mathfrak{F}$ is in $\mathfrak{H}$ contained, does not imply a corresponding inclusion between their projectors. For example, in $sym(4)$, an $\mathfrak{\ddot{U}}$-projector cannot be contained in an $\mathfrak{NA}$-projector, although, $\mathfrak{\ddot{U}}\subseteq \mathfrak{NA}$.
Let $\mathfrak{F}$ and $\mathfrak{H}$ be saturated formations. We say that $\mathfrak{F}$ is strongly contained in $\mathfrak{H}$ if, in each finite soluble group $G$ a $\mathfrak{F}$-projector of $G$ is contained, as subgroup, in a $\mathfrak{H}$-projector of $G$. To denote it we write $\mathfrak{F} \ll\mathfrak{H}$. It is clear that $(\mathscr{F} ,\ll)$ is a partial order.
The strong containment for Schunck classes is quite well understood. But in the context of the family, $\mathscr{F}$ of saturated formations in $\mathfrak{S}$ the theory is less satisfactory. The proofs are very complicated, and results of general applicability are difficult to find.
In this talk, we present a small historical tour through the strong containment relationship, its challenges and new consideration of this relation in another universe. (In collaboration with Anselmo Torresblanca Badillo)
Friday, August 9
14.30-15.30
Lev Glebsky (Universidad Autonoma de San Luis Potosi, México)
Approximations of groups and stability
In the talk I plan to present several definitions of approximations of groups, starting with one of A. Turing, discuss motivations, results and open problems. Then I will speak about stability (with respect to some asymptotic homomorphisms). The stability may serve as an obstruction for approximability. Unfortunately, the known results on stability is not sufficient to prove non-approximabilty results in most interesting cases.
15.40- 16.10
Noraí Rocco (University of Brasilia, Brazil)
Some Bounds for the Orders of Non-abelian Tensor Products of Groups
Let $G$ and $H$ be groups that act compatibly on each other. In this talk we consider the group $\eta(G,H)$, a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Our purpose is to discuss the influence of the finiteness of the set of all tensors $T_{\otimes} (G, H) := \{ g \otimes h : g \in G, \, h \in H \}$ on the finiteness of the non-abelian tensor product $G \otimes H$. We prove that if $|T_{\otimes} (G, H)| = m,$ then $G \otimes H$ is finite with $m$-bounded order. We also address some results concerning finiteness conditions for the non-abelian tensor square of a group.
(This is a joint work with Raimundo Bastos and Irene Nakaoka).
16.10-16-40
Liudmila Sabinina (UAEM, Cuernavaca, Mexico)
On automorphic Moufang loops
The variety of Moufang loops and the variety of commutative Moufang loops are the most known varieties of loops. In our previous work it was shown that the variety of automorphic Moufang loops, i.e Moufang loops with some additional family of automorphisms, is a join of the variety of groups and the variety of commutative Moufang loops. In my talk I will present the variety of automorphic Moufang loops invariant under isotopy and discuss some related questions. It will be shown that any loop isotopic to 3-generator automorphic Moufang loop is again automorphic.
(This is a joint work with Alexandre Grishkov, Ricardo Diaz and Marina Rasskazova).
17.10-18.10
Eli Aljadeff (Technion-Israel)
Group gradings on finite dimensional division algebras
Finite group grading play a key role in the study of finite dimensional division algebras and more generally in the study of finite dimensional central simple algebras. For instance, crossed product algebras, which provide the bridge between Brauer groups and Galois cohomology, and symbol algebras, which provide the bridge between Brauer groups and $K$-theory, are both naturally graded algebras.
We consider the following question: What are all possible (finite) group gradings on finite dimensional $k$-central division algebras?
This seems to be a difficult problem in its full generality. For example we do not know for which finite groups $G$, there exists a field $k_{0}$ and a $2$-cocycle $\alpha \in H^{2}(G, k_{0}^{*})$ such that the twisted group algebra $k_{0}^{\alpha}G$ is a division algebra. In this talk we give, by means of generic constructions, a complete answer in the case where the center $k$ contains an algebraically closed field of characteristic zero.
18.20-18.50
Rosemary Miguel Pires (Fluminense Federal University, Brazil)
Groups with triality associated to code loops
The study of code loops began with the paper Code loops of Griess. The complete classification of all code loops up to isomorphism is very difficult, since the number of those loops grows very quickly with growth of rank. Now we have classification of all code loops of rank up to $8$. By the other hand, using the concept of the characteristic vectors associated with code loops, the classification of all code loops of rank $n< 5$ also has been done: there are exactly 5 nonassociative code loops of rank $3$ (up to isomorphism) and 16 nonassociative code loops of rank $4$ (up to isomorphism).
A group $G$ that admits an action of $S_{3} = \left\{\sigma, \rho \ | \ \sigma^{2} = \rho^{3} = (\sigma \rho)^{2} = 1\right\}$ satisfying $(x^{-1}x^{\sigma})(x^{-1}x^{\sigma})^{\rho}(x^{-1}x^{\sigma})^{\rho^{2}} = 1$ is called a group with triality. Groups with triality are naturally connected with Moufang loops. In the paper Groups with triality, Grishkov and Zavarnitsine describe all possible groups with triality associated with a given Moufang loop. In this talk, with a computational approach, we present some results obtained applying the tecniques developed by Grishkov and Zavarnitsine to the code loops of rank $3$.
(This is a joint work with Alexandre Grishkov (University of São Paulo, Brazil).