Studies in fuzzy groups

Abstract

In this thesis we first extend the notion of fuzzy normality to the notion of normality of a fuzzy subgroup in another fuzzy group. This leads to the study of normal series of fuzzy subgroups, and this study includes solvable and nilpotent fuzzy groups, and the fuzzy version of the Jordan-Hõlder Theorem. Furthermore we use the notion of normality to study products and direct products of fuzzy subgroups. We present a notion of fuzzy isomorphism which enables us to state and prove the three well-known isomorphism theorems and the fact that the internal direct product of two normal fuzzy subgroups is isomorphic to the external direct product of the same fuzzy subgroups. A brief discussion on fuzzy subgroups generated by fuzzy subsets is also presented, and this leads to the fuzzy version of the Basis Theorem. Finally, the notion of direct product enables us to study decomposable and indecomposable fuzzy subgroups, and this study includes the fuzzy version of the Remak-Krull-Schmidt Theorem.