Studies in fuzzy groups

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

The classification of fuzzy groups of finite cyclic groups Zpn Zqm Zr and Zp1 Zp2 Zpn for distinct prime numbers p; q; r; p1; p2; ; pn and n;m 2 Z+

Let G be the cyclic group Zpn _ Zqm _ Zr where p; q; r are distinct primes and n;m 2 Z+. Using the criss-cut method by Murali and Makamba, we determine in general the number of distinct fuzzy subgroups of G. This is achieved by using the maximal chains of subgroups of the respective groups, and the equivalence relation given in their research papers. For cases of m, the number of fuzzy subgroups is _rst given, from which the general pattern for G is achieved. Murali and Makamba discussed the number of fuzzy subgroups of Zpn _ Zqm using the cross-cut method. A brief revisit of the group Zpn _Zqm is done using the criss-cut method. The formulae for _nding the number of distinct fuzzy subgroups in each of the cases is given and proofs provided. Furthermore, we classify the fuzzy subgroups of the group Zp1_Zp2__ _ __Zpn for p1; p2; _ _ _ ; pn distinct primes and n 2 Z+ using the criss-cut method. An algorithm for counting the distinct fuzzy subgroups of this group is developed