The construction of fuzzy subgroups of groups with units modulo 20 and 21

The number of fuzzy subgroups (NFSGs) of the groups of units U20 and U21 are determining in this study. The (NFSGs) of U20 was found based on diagram and using the lemma which state that the (NFSGs) of G is equal to the number of chains (NC) on the subgroups lattice of G. Moreover, we explain that there are only two fuzzy subgroups for a prime order group. Lagrange’s theorem is more important in this work, from Lagrange’s theorem, there is no nontrivial subgroup of U21 since the order of the group is prime. We prove that, if P1 (μ) = U21, then we get μ1(x) = θ1, ∀x ∈ U21. Thus we have one fuzzy subgroup of P1 (μ) = U21. If P1 (μ) = {1}, then we obtain μ2(x) = θ1, ∀x ∈ {1} and μ2 (x) = θ2,∀x∈U21\{1}. Thus we have 2 fuzzy subgroups of U21. We came up with the numbers 21 and 2 respectively.