• Study of semigroups, monoids, automata, and formal languages.

In computer science, there has been 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.

• Arithmetic and structural study of factored groups. Structural study of ties.

When considering a factored group G=AB as a product of two subgroups, related by 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 the properties of G. Ties are associated with three-factored groups with structural properties that effectively determine solutions to the quantum Yang-Baxter equation.

• Group actions. 

Certain types of groups are defined by the actions of the groups on main factors or other normal sections. Of particular importance are subgroups that cover or avoid all of the group's main factors, as well as actions that determine specific constraints.

• Analysis of 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 in this direction.

• Study of the normal and permutable structure of certain families of groups with finiteness conditions. 

In recent years, there has been interest in groups where all subnormal subgroups are normal, permutable, or Sylow-permutable, both in terms of vanishing groups and extensions to classes of infinite groups. We also develop computational techniques to study these groups with GAP.