Research Projects

This project aims to understand the z-classes of linear algebraic groups over an arbitrary field, focusing on solvable algebraic groups defined over k. The problem of z-classes is well understood for reductive algebraic groups defined over a field of type (F), but not for solvable algebraic groups defined over k. The project has been studied for the Borel subgroup of the general linear group over any infinite field k, proving that the number of z-classes in the group of n by n upper triangular matrices is finite when n is less than 6, and infinite when n is greater than or equal to 6 provided the field is infinite. The main theme of the project is to continue studying nilpotent algebraic groups over an algebraically closed field, nilpotent algebraic groups defined over an arbitrary field, and solvable algebraic groups defined over k. The project also aims to study finite groups of Lie type and determine the number of z-classes in the symplectic group and orthogonal group.