Cai Heng Li, Chair Professor from the Department of Mathematics at the Southern University of Science and Technology (SUSTech), has made breakthrough progress on the problem of factorizing simple groups. The result consists of three research articles, published in top mathematical journals. These were entitled “Factorizations of almost simple groups with a factor having many nonsolvable composition factors” in the Journal of Algebra, “Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups” in Memoirs of the American Mathematical Society, and “On solvable factors of almost simple groups” in Advances in Mathematics.
The concept of a “group” occurs in various branches of mathematics as a precise measure of the degree of symmetry of an object. “Group Theory” is a branch of mathematics studying symmetry. Symmetry appears everywhere and keeps the balance and stability of the world. Objects with high symmetry have a simple and beautiful property and play important roles in the activities for human beings to understand the world. Group Theory provides fundamental tools for various branches of mathematics and has many applications in Physic, Chemistry, group-based cryptography, the graph isomorphism problem (related to computer science and chemical documentation), the mixing rate of Markov chains, and the design of interconnection networks for large interacting arrays of processors in parallel computations.
“Simple groups” are basic building blocks for general groups, just as atoms to matter. During several decades last century, classifying finite simple groups was one of the central problems in mathematics. The problem was solved in the 1980s by the work of hundreds of mathematicians. Then classifying factorizations of simple groups has been an important problem in algebra, especially in group theory, which naturally relates to many problems in other mathematics areas and has many applications.
Studying this problem has been an active research topic during the past 40 years, and many first-class experts in group theory have made significant contributions to the problem. However, the problem is still unsettled and challenging. Recently, the above-mentioned result of Cai Heng Li and his collaborators solved two critical parts of this problem. They solved the problem for cases where one factor has at least two non-solvable composition factors or one factor is solvable. This reduces the problem to the case where both factors have exactly one non-solvable composition factor and shapes a new approach to tackle the half-century open problem.
Along with the new approach, Cai Heng Li and his collaborators are working towards a complete solution for the problem. The research is partially supported by the National Natural Science Foundation of China (NSFC) research grants.