Length functions and fuzzy groups

Abstract

Fuzzy sets were introduced by Zadeh [7] in 1965 and fuzzy groups by Rosenfeld [4] in 1971. Fuzzy groups were also considered by Anthony and Sherwood [I] in 1979 and in 1981 the theory was developed by Das [2]. As in the paper of Das [2] we also show how a fuzzy subgroup of a group G is determined by a chain of subgroups of G. However, we wish to point out that fuzzy subgroups are constructed in a way which is analogous to that of constructing certain length functions of G (Wilkens [S, 61). Thus in Section 3 we consider the theory of length functions and we show the connection between certain length functions and fuzzy subgroups of a group. In Section 4 we apply the structural results of Section 3 on finite solvable groups and we gain a generalization of a theorem of Das [2]. Our notation is standard. However, we use the symbol e to denote the identity element of a group G and I denotes a real number. We start with a preliminary section where we briefly analyse the definition of a fuzzy subgroup given by Rosenfeld [4] (naturally we use this definition of a fuzzy subgroup throughout the paper).