# Fuzzy Group Theory

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 182)

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 182)

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.

Abelian group Abtract Recognition Problems Fuzzy Fuzzy Group Theory Fuzzy Groups Fuzzy Semigroups Group theory coding theory cognition construction linear optimization

- DOI https://doi.org/10.1007/b12359
- Copyright Information Springer-Verlag Berlin/Heidelberg 2005
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Engineering
- Print ISBN 978-3-540-25072-2
- Online ISBN 978-3-540-32395-2
- Series Print ISSN 1434-9922
- Series Online ISSN 1860-0808
- About this book