The fuzzy subgroups results involving multiple sums

TL;DR

This paper develops formulas to classify fuzzy subgroups of nilpotent groups formed from Cartesian products of p-groups, focusing on dihedral and cyclic groups of specific orders.

Contribution

It introduces explicit formulas for counting fuzzy subgroups of Cartesian products of p-groups, especially dihedral and cyclic groups, expanding prior classifications.

Findings

Derived explicit formulas for fuzzy subgroup counts

Classified fuzzy subgroups of Cartesian products of p-groups

Extended understanding of fuzzy subgroup structures in nilpotent groups

Abstract

The theory of fuzzy sets has a wide range of applications, one of which is that of fuzzy groups . The fuzzy sets were introduced by Zadeh. Even though, the story of fuzzy logic started much earlier, it was specially designed mathematically to represent uncertainty and vagueness. It was also, to provide formalized tools for dealing with the imprecision intrinsic to many problems. A group is said to be nilpotent if it has a normal series of a finite length n. By this notion, every finite p-group is nilpotent. Nilpotent structures such as the p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian product (otherwise known as the product set) plays vital roles in the course of synthesizing the abstract groups. Previous studies have determined the number of distinct fuzzy subgroups of various finite p-groups including those of square-free order. However, not much work has been done on the fuzzy subgroup classification for the nilpotent groups formed from the Cartesian products of p-groups through their computations. This work is therefore designed to classify the nilpotent groups formed from the Cartesian products of p-groups through their computations. In this paper, the Cartesian products of p-groups were taken to obtain nilpotent groups. the explicit formulae is given for the number of distinct fuzzy subgroups of the Cartesian product of the dihedral group of order eight with a cyclic group of order of an n power of two for, which n is not less than three