FUZZY NORMAL-SUBGROUPS AND FUZZY QUOTIENTS

In case of groups, a normal subgroup N of a group G can be defined with three equivalent approaches, namely (i) as a kernel of some group homomorphism, (ii) as a subgroup commuting with every element of G, and (iii) as the subgroup N closed with respect to conjugates of elements of N. In case of generalizations of this concept to fuzzy sets, the elementwise third approach is found more suitable and hence is widely adopted by many authors. In this paper we further study this concept of fuzzy normal subgroups and introduce fuzzy quotients called alpha-fuzzy quotient groups for all alpha is-an-element-of [0, 1]. Two different definitions of these alpha-fuzzy quotients have been given which are shown to lead to isomorphic structures. We also study the direct products of fuzzy (normal) subgroups (min-product as per the terminology of Sherwood (1983)) and the problem of writing a fuzzy (normal) subgroup of a direct product of some groups as a direct product of certain fuzzy (normal) subgroups, has been discussed in detail.