Burnside matrices and subgroup embeddings in finite groups

We apply the technique of Burnside matrices (tables of marks) to recognition of subgroup embeddings in a finite group. This tool allowed us to work out efficient tests for such embedding properties as abnormality, pronormality, paranormality, their weak analogies (weak abnormality, extended Frattini argument, polynormality), the subnormalizer condition and weak normality in the sense of K. H. Muller. Two combinatorial functions on the subgroup lattice of a finite group are considered. Both of them may be computed with the help of the respective table of marks. The algorithms are implemented within the computer algebra package CAP, version 4.1. Our codes make use of the library TOM there and fulfil calculations with integers instead of computing with group elements. Three long-standing problems in the area of subgroup embeddings were solved with the help of these programs. The respective counterexamples are described. In addition to the plenary talk the author would Like to demonstrate his codes.