Nilpotent Fuzzy Subgroups

Elaheh Mohammadzadeh,Rajab Borzooei

doi:10.3390/math6020027

Abstract

In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization of abstract nilpotent groups. For this, we show that a group G is nilpotent if and only if any fuzzy subgroup of G is a g-nilpotent fuzzy subgroup of G. In particular, we construct a nilpotent group via a g-nilpotent fuzzy subgroup. Finally, we characterize the elements of any maximal normal abelian subgroup by using a g-nilpotent fuzzy subgroup.

Highlights

Applying the concept of fuzzy sets of Zadeh [1] to group theory, Rosenfeld [2] introduced the notion of a fuzzy subgroup as early as 1971
[8] Guptaa and Sarmahas, defined the commutator of a pair of fuzzy subsets of a group to generate the descending central chain of fuzzy subgroups of a given fuzzy subgroup and they proposed a new definition of a nilpotent fuzzy subgroup through its descending central chain
By the notion of a g-nilpotent fuzzy subgroup we can investigate on fuzzification of nilpotent groups

Summary

Introduction

Applying the concept of fuzzy sets of Zadeh [1] to group theory, Rosenfeld [2] introduced the notion of a fuzzy subgroup as early as 1971. In [7] Kim proposed the notion of a nilpotent fuzzy subgroup There, he attached to a fuzzy subgroup an ascending series of subgroups of the underlying group to define nilpotency of the fuzzy subgroup. In [8] Guptaa and Sarmahas, defined the commutator of a pair of fuzzy subsets of a group to generate the descending central chain of fuzzy subgroups of a given fuzzy subgroup and they proposed a new definition of a nilpotent fuzzy subgroup through its descending central chain. They proved that every Abelian (see [9]) fuzzy subgroup is nilpotent. We prove that a finite maximal normal subgroup can control the g-nilpotent fuzzy subgroup and makes it finite

Preliminary

Good Nilpotent Fuzzy Subgroups

Conclusions