Classification of Distinct Fuzzy Subgroups of the Dihedral Group Dp nq for p and q distinct primes and n ∈ N

In this dissertation, we classify distinct fuzzy subgroups of the dihedral group Dpnq, for p and q distinct primes and n ∈ N, under a natural equivalence relation of fuzzy subgroups and a fuzzy isomorphism. We aim to present formulae for the number of maximal chains and the number of distinct fuzzy subgroups of this group. Our study will include some theory on non-abelian groups since the classification of distinct fuzzy subgroups of this group relies on the crisp characterization of maximal chains. We give the definition of a natural equivalence relation introduced by Murali and Makamba in [67] which we will use in this study. Based on this definition, we introduce two counting techniques that we will use to compute the number of distinct fuzzy subgroups of Dpnq. In this dissertation, we use the criss-cut counting technique as our primary method of enumeration, and the cross-cut method serves as a means of verifying results we obtain from our primary method. To classify distinct fuzzy subgroups of this group, we begin by investigating the dihedral groups Dpnq, for p and q distinct primes and specific values of n = 2 and 3 to observe a trend. We classify the flags of these groups using the characterization of flags introduced in [93]. From this characterization, we then present formulae for the number of distinct fuzzy subgroups attributed to the flags of Dp 2q and Dp 3q . To generalise results for Dpnq, for p and q distinct primes and n ∈ N, we characterize the flags of this group and classify them as either cyclic, mdcyclic for 1 ≤ m ≤ n, or b-cyclic. Finally, we establish a general formula for the number of distinct fuzzy subgroups obtainable from these flags. We conclude by comparing results obtained from using our general formula to those obtained by other researchers for the same group. Based on the results from this study, we give an outline of future research wor