On the number of fuzzy subgroups of symmetric group $ S_5 $

TL;DR

This paper calculates the exact number of distinct fuzzy subgroups of the symmetric group S_5 using the Inclusion-Exclusion principle, providing new insights into fuzzy subgroup enumeration.

Contribution

It introduces an equivalence relation on fuzzy subgroups and applies the Inclusion-Exclusion principle to determine the count for S_5, advancing fuzzy subgroup theory.

Findings

Number of fuzzy subgroups of S_5 computed

Established inequalities for the count for n ≥ 5

Defined an equivalence relation to classify fuzzy subgroups

Abstract

This article computes the number of fuzzy subgroups of symmetric group $S_5$. First, an equivalence relation on the set of all fuzzy subgroups of a group G is defined.Without any equivalence relation on fuzzy subgroups of group G, the number of fuzzy subgroups is infinite, even for the trivial group. The\emph{Inclusion-Exclusion principle} is used to determine the number of distinct fuzzy subgroups of \textbf{symmetric group} $S_5$. Some inequalities satisfied by this number are also established for $n \geq 5$.