Research Group on Arithmetic and Structural Properties of Groups, Applications I

Description

A natural line of research in the field of group theory is the study of arithmetic and structural properties of groups, in which this research group has consolidated experience of more than fifteen years. The techniques of group class theory and its representations are fundamental to this study. These techniques can also be used for the study of structural problems of semigroups, based on the already existing study of interactions between groups and formal languages and automata, as well as interactions between trifactorised groups, group actions, fathoms and the Yang-Baxter equation.

This group aims to advance knowledge of:

Factor groups. Structural study of the fathoms and their relation to the Yang-Baxter equation.
Group actions on certain normal subgroups and on their main factors.
Structural influence of the relationships between different families of subgroups and their immersion properties.
The normal and permutable structure of certain families of groups with finiteness conditions.
The role of groups in semi-groups and their representations. Formal languages and automata.

This group works in coordination with other teams based at the University of Zaragoza and the Public University of Navarra, on the one hand, and at the Universitat Politècnica de València, on the other. Keywords: group, permutability, group actions, semigroup, formal language, automaton, fathom, immersion property.

Goals CT

Arithmetic and structural study of factor groups.
Structural study of flanges and their influence on the study of the Yang-Baxter equation.
Study of the actions of defunct groups on their main factors.
Study of the structural impact of immersion properties of distinguished families of subgroups.
Study of the normal and permutable structure of certain families of groups with finiteness conditions.
Study of semigroups, monoids, automata and formal languages.

Research lines

Group actions
Certain classes of groups are defined by the actions of the groups with regard to the main factors or other normal sections. Special importance is attached to subgroups that cover or avoid all the main factors of the group, as well as actions that determine flanges of special type.
Analysis on the structural impact of immersion properties of distinguished families of subgroups
A natural problem in group theory is: what can we say about a group in which all subgroups of a relevant family of subgroups satisfy a certain property? We intend to make contributions along these lines.
Study of the normal and permutable structure of certain families of groups with finiteness conditions
In recent years, groups where all subnormal subgroups are normal, permutable, or Sylow-permutable have been of interest, both in terms of finite groups and extensions to classes of infinite groups. We have also developed computational techniques to study these groups with GAP.
Arithmetic and structural study of factor groups. Structural study of fathoms
When a factor group G=AB is considered as a product of two subgroups, related with certain permutability conditions, the natural question is to determine what we can say about G from the properties of A and B, and what we can say about A and B from properties of G.

The fathoms are associated to trifactor groups with structural properties that effectively determine solutions of the quantum Yang-Baxter equation.
Study of semigroups, monoids, automata and formal languages
In the field of computer science, there has been a growing interest in the study of semigroups and monoids in relation to automata and formal languages. We aim to apply techniques from group theory and universal algebra to the analysis of these objects.