Fuzzy Subsets and Fuzzy Subgroups -- Fuzzy Caley's Theorem and Fuzzy Lagrange's Theorem -- Nilpotent, Commutator, and Solvable Fuzzy Subgroups -- Characterization of Certain Groups and Fuzzy Subgroups -- Free Fuzzy Subgroups and Fuzzy Subgroup Presentations -- Fuzzy Subgroups of Abelian Groups -- Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups -- Equivalence of Fuzzy Subgroups of Finite Abelian Groups -- Lattices of Fuzzy Subgroups -- Membership Functions From Similarity Relations.This book presents an up-to-date account of research in important topics of fuzzy group theory. The book concentrates on the theoretical aspects of fuzzy subgroups of a group. It also includes applications to some abstract recognition problems and to coding theory. The book begins with basic properties of fuzzy subgroups. The notions of ascending series and descending series of fuzzy subgroups are used to define nilpotency of a fuzzy subgroup. Fuzzy subgroups of Hamiltonian, solvable, P-Hall, and nilpotent groups are discussed. Construction of free fuzzy subgroups is determined. Numerical invariants of fuzzy subgroups of Abelian groups are developed. The problem in group theory of obtaining conditions under which a group can be expressed as a direct product of its normal subgroups is considered. The number of fuzzy subgroups (up to an equivalence relation) of certain finite Abelian groups is determined. Methods for deriving fuzzy theorems from crisp ones are presented and the embedding of lattices of fuzzy subgroups into lattices of crisp groups is discussed. Deriving membership functions from similarity relations is considered. The material presented in this book makes it a good reference for graduate students and researchers working in fuzzy group theory.