Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups

Abstract

Abstract We define a notion of ‘containment’ of an ordinary kernel of a group homomorphism in a fuzzy subgroup. Using this idea, we provide the long-awaited solution of the problem of showing a one-to-one correspondence between the family of fuzzy subgroups of a group, containing the kernel of a given homomorphism, and the family of fuzzy subgroups of the homomorphic image of the given group. It is shown that an ordinary kernel gives rise to the notion of fuzzy quotient group in a natural way. Consequently, the fundamental theorem of homomorphisms is established for fuzzy subgroups. Moreover, we provide new proofs for the facts, that the homomorphic image of a fuzzy subgroup is always a fuzzy subgroup, and fuzzy normality is invariant under surjective homomorphism.