Algebra Research at Lincoln
Group Theory is one of the most important parts of modern Algebra. Groups can be regarded as measures of symmetry understood in the most broad sense as invariants under "admissible" transformations, specific to each subject area. The language of group theory is therefore used throughout mathematics, as well as in natural sciences. Abstract Group Theory studies the intrinsic structure of groups, which are quite fascinating objects from the pure mathematical point of view.
One of the difficulties in the study of groups is that the group operation is not commutative, and groups in general do not have any linear structure. But certain classes of groups are closer to being commutative than others. Special methods are used to construct linear structures from the group, such as a vector space, or a Lie ring. This makes it possible to apply the technique of linear algebra: for example, an automorphism of a group can sometimes be regarded as a linear transformation, for which there arise eigenspaces, Jordan normal form, etc. This illuminates the group in a new "linear" light, and a problem becomes easier to handle.
In the theory of finite groups, many problems nowadays can be reduced to soluble and nilpotent groups by using the classification of finite simple groups. Further study of soluble groups based on representation theory provides further reductions to nilpotent groups. The latter in turn are often studied by using Lie ring methods.
One of the directions of our research in Algebra is the study of automorphisms with restrictions on their fixed points, with the aim of achieving greater commutativity of the group, or of a suitable large subgroup. Another strongpoint is developing novel Lie ring methods, as well as proving results on Lie rings and their automorphisms, which can be applied to groups.
Much of the group's research is conducted in international collaboration. The Brazilian research council CNPq provided a Special Visiting Researcher grant for Prof Evgeny Khukhro, who regularly visits University of Brasilia for joint research on application of Hall—Higman—type theorems and Lie ring methods, as well as developing new length-type parameters for nonsoluble finite groups with applications to profinite groups. Collaboration with Universities of Ankara and Dogus University in Istanbul resulted in new Hall—Higman—type results for Frobenius—type groups of automorphisms. In several joint papers with researchers in Novosibirsk (Russia), Mulhouse (France), and Brasilia, Lie ring methods were successfully applied for studying nilpotent groups and their automorphisms.