Algebraic Graph Theory Seminar - Róbert Jajcay (26.4.2024)
Friday 3.5.2024 at 13:00, Lecture room M/IX (online too)
Róbert Jajcay:
From Orientably Regular Maps to Cyclic Complementary Extensions of Finite Groups
Abstract:
Orientably regular maps are defined via the existence of a group of orientation preserving automorphisms acting regularly on the set of darts of the map. It is therefore perhaps not surprising that the study of orientably regular maps led to insights in finite permutation group theory. The concept of a skew-morphism - developed in the context of regular Cayley maps and introduced more than 20 years ago - constitutes one such example bridging topological graph theory and cyclic complementary extensions of groups. While automorphism groups of regular Cayley maps can be easily seen to be cyclic complementary extensions of their underlying groups via special structure preserving skew-morphisms, it is a bit more surprising that all skew-morphisms give rise to cyclic complementary extensions; which became known under the name of skew-products. It is even more surprising that all cyclic complementary extensions give rise to a skew-morphism. Nevertheless, not all cyclic complementary extensions are skew-products which left a gap in our understanding of cyclic complementary extensions of finite groups.
Recently, together with Kan Hu, we have been able to fill this gap by introducing a generalization of the power function of a skew-morphism which we call an extended power function, and by finding a universal construction of cyclic complementary extensions of groups via skew-morphisms and their extended power functions. We have shown that all cyclic complementary extensions are constructed in this way which means that in order to classify cyclic complementary extensions of a specific finite group it suffices to classify its skew-morphisms and their extended power functions. In view of this, we started investigating extended power functions of special classes of skew-morphisms, and completed the classification of extended power functions of group automorphisms of cyclic groups.
Those of you who are not able to attend in person or who are still uncertain about the safety of attending in person are welcome to attend via MS Teams. In either case, we hope to see as many of you as possible (either in person or virtually) at our Friday gatherings.