Some New Directions in Commutativity Degree of Finite Groups: Recent Developments

Classification of finite groups is a central problem in theory of groups. Even though finite abelian groups have been completely classified, a lot still remains to be done as far as non-abelian groups are concerned. People all over the world have used various types of invariants for classifying finite non-abelian groups. The commutativity degree of a finite group is one such invariant. It is the probability that two randomly chosen group elements commute. This book contains some new results on commutativity degree of finite groups. Several generalizations of the concept of commutativity degree are introduced and studied in this book. In particular, some characterizations of finite groups are obtained in terms of commutativity degree and generalized commutativity degrees. Some generalizations of a classical result of F. G. Frobenius regarding the number of solutions of a commutator equation in a finite group are also disscussed. The book begins with a brief recollection of a few definitions and well-known results from several relevant topics, which constitute the minimum prerequisites for the subsequent chapters, and ends with a discussion on some possible research problems.