| In this thesis, we carry out a deep and detailed investigation into (λ,μ) fuzzy groups. The similar results to those in the sense of Rosenfeld and Bhakat-Das fuzzy groups are obtained. As a consequence of our research, the theory of (λ,μ) fuzzy group is complemented and some results in the crisp and fuzzy group are extended. The main research is summarized as follows.Firstly, we recall some basic notions such as fuzzy set, fuzzy subgroup etc. and related results employed in the thesis. Then, we review the current study on (λ,μ) fuzzy subgroups.Secondly, we give some fundamental properties of (λ,μ) fuzzy subgroups, their level subgroups, and the level subgroup chain of a(λ,μ) fuzzy subgroup. The relationship between a (λ,μ) fuzzy subgroup of a group and its level subgroups is investigated. Moreover, we deal with the issue of the expressability of (λ,μ) fuzzy subgroups as the union of two (λ,μ) fuzzy subgroups. At last, a method is proposed to construct the (λ,μ) fuzzy subgroup generated by a given fuzzy subset.Thirdly, we study the basic properties of (λ,μ) fuzzy normal subgroups, and present some results related to (λ,μ) fuzzy cosets and (λ,μ) fuzzy quotient subgroups. We prove that there is a natural one-to-one correspondence between the (λ,μ) fuzzy right cosets of a (λ,μ) fuzzy subgroup A of a group G and the right cosets of the subgroup Aμ of G.Finally, we introduce the notions of (λ,μ) fuzzy characteristic subgroup and (λ,μ) fuzzy conjugate subgroup, discuss their fundamental properties and obtain the condition under which a (λ,μ) fuzzy subgroup satisfies the properties fulfilled by fuzzy characteristic subgroups. We also reveal the relationship between a (λ,μ) characteristic subgroup and a(λ,μ) fuzzy conjugate subgroup and their level subgroups and give the characterization of a (λ,μ) fuzzy characteristic subgroups by their level subgroups. |
