Principle,Maximal subgroups.

The aspect of pure Mathematics has undergone a lot of dynamic developments over the years . Concerning the theory of fuzzy group , the classification, most especially the finite p-groups cannot be overlooked. For instance, many researchers have treated cases of finite abelian groups. Since inception , the study has been extended to some other important classes of finite abelian and nonabelian groups such as the dihedral , quaternion, semidihedral, and hamiltonian groups. Other different approaches have been so far, applied for the classification. The Fuzzy sets were introduced by Zadeh in 1965. Even though, the story of Fuzzy logic started much more 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. The term fuzzy logic is generic as it can be used to describe the likes of fuzzy arithmetic, fuzzy mathematical programming, fuzzy topology, fuzzy graph theory ad fuzzy data analysis which are customarily called fuzzy set theory. This theory of fuzzy sets has a wide range of applications, one of which is that of fuzzy groups developed by Rosenfield in 1971. This by far, plays a pioneering role for the study of fuzzy algebraic structures. Other notions have been developed based on this theory. These, amongst others, include the notion of level subgroups by P.S. Das used to characterize fuzzy subgroups of finite groups and that of equivalence of fuzzy subgroups introduced by Murali and Makamba which we use in this work. ( Please, see [ 1 - 9 ] ) By the way, A group is nilpotent if it has a normal series of a finite length n.

G = G0 ≥ G1 ≥ G2 ≥ · · · ≥ Gn = {e}, where Gi/Gi+1 ≤ Z(G/Gi+1).

By this notion, every finite p-group is nilpotent. The nilpotence property is an hereditary one. Thus,

(i) Any finite product of nilpotent group is nilpotent.

(ii) If G is nilpotent of a class c, then, every subgroup and quotient group of G is nilpotent and of class ≤ c.

The problem of classifying the fuzzy subgroups of a finite group has so far experienced a very rapid progress. One particular case or the other have been treated by several papers such as the finite abelian as well as the non-abelian groups. The number of distinct fuzzy subgroups of a finite cyclic group of square-free order has been determined. Moreover, a recurrence relation is indicated which can successfully be used to count the number of distinct fuzzy subgroups for two classes of finite abelian groups. They are the arbitrary finite cyclic groups and finite elementary abelian p-groups. For the first class, the explicit formula obtained gave rise to an expression of a well-known central Delannoy numbers. Some forms of propositions for classifying fuzzy subgroups for a class of finite p-groups have been made by Marius Tarnauceaus. It was from there, the study was extended to some important classes of finite non-abelian groups such as the dihedral and hamiltonian groups. And thus, a method of determining the number and nature of fuzzy subgroups was developed with respect to the equivalence relation. There are other different approaches for the classification. The corresponding equivalence classes of fuzzy subgroups are closely connected to the chains of subgroups, and an essential role in solving counting problem is again played by the inclusion - exclusion principle. This hereby leads to some recurrence relations, whose solutions have been easily found. For the purpose of using the Inclusion - Exclusion principle for generating the number of fuzzy subgroups, the finite p-groups has to be explored up to the maximal subgroups. The responsibility of describing the fuzzy subgroup structure of the finite nilpotent groups is the desired objective of this work. Suppose that (G, ·, e) is a group with identity e. Let S(G) denote the collection of all fuzzy subsets of G. An element λ ∈ S(G) is called a fuzzy subgroup of G whenever it satisfies some certain given conditions . Such conditions are as follows :

(i) λ(ab) ≥∈ {λ(a), λ(b)}, ∀ a, b ∈ G;

(ii) λ(a ≥ λ(a) for any a ∈ G.

−1 −1 −1

And, since (a ) = a, we have that λ(a ) = λ(a), for any a ∈ G.

Also, by this notation and definition, λ(e) = sup λ(G). [Marius [6]].

Theorem :The set F L(G) possessing all fuzzy subgroups of G forms a lattice under the usual ordering of fuzzy set inclusion. This is called the fuzzy subgroup lattice of G.

We define the level subset:

λGβ = {a ∈ G/λ(a) ≥ β} for each β ∈ [0, 1] The fuzzy subgroups of a finite p-group G are thus, characterized, based on these subsets. In the sequel, λ is a fuzzy subgroup of G if and only if its level subsets are subgroups in G. This theorem gives a link between F L(G) and L(G), the classical subgroup lattice of G.F Moreover, some natural relations on S(G) can also be used in the process of classifying the fuzzy subgroups of a finite q-group G. One of them is defined by: λ ∼ γ iff (λ(a) > λ(b) ⇐⇒ v(a) > v(b), ∀ a, b ∈ G). Alos, two fuzzy subgroups λ, γ of G and said to be distinct if λ × v.

As a result of this development, let G be a finite p-group and suppose that λ : G −→ [0, 1] is a fuzzy subgroup of G. Put λ(G) = {β1, β2, . . . , βk} with the assumption that β1 < β2 > · · · > βk. Then, ends in G is determined by λ.

λGβ ⊂ λGβ ⊂ · · · ⊂ λGβ = G (a)

1 2 k

Also, we have that:

λ(a) = βt ⇐⇒ t = max{r/a ∈ λGβ } ⇐⇒ a ∈ λGβ \λGβ ,

r t t−1

for any a ∈ G and t = 1, . . . , k, where by convention, set λGβ = ϕ.

We are going to adopt a method that will be used in counting the chains of fuzzy subgroups of an arbitrary finite p-group G is described. Suppose that M1, M2, . . . , Mt are the maximal subgroups of G, and denote by h(G) the number of chains of subgroups of G which ends in G. By simply applying the technique of computing h(G), using the application of the Inclusion-Exclusion Principle, we have that:

 !!

X X \  t−1 h(G) = 2 h(Mr) − h(Mr ∩ Mr ) + · · · + (−1) h Mr (#)

r=1 1≤r1<r2≤t r=1

In [6], (#) was used to obtain the explicit formulas for some positive integers n.

Theorem [ 1 ] [Marius]: The number of distinct fuzzy subgroups of a finite p-group of order p which have a cyclic maximal subgroup is:

n n−1

(i) h(Zpn ) = 2 , (ii) h(Zp × Z n−1 ) = 2 [2 + (n − 1)p]

3 The distinct Number of The Fuzzy Subgroups of The Nilpotent Group of (D 3 × C2m ) For m ≥ 3

Proposition 1 ( see [ 13 ] ) : Suppose that G = Z4 × Z2n , n ≥ 2. Then, h(G) = 2 [n + 5n − 2]

Proof : G has three maximal subgroups of which two are isomorphic to Z2 × Z2n and the third is isomorphic to Z4 × Z2n−1 .

Hence, h(Z4 × Z2n ) = 2h(Z2 × Z2n ) + 2 h(Z2 × Z n−1 ) + 2 h(Z2 × Z n−2 )

3 4 n−2

+ 2 h(Z2 × Z n−3 ) + 2 h(Z2 × Z n−4 ) + · · · + 2 h(Z2 × Z22 )

= 2 [2(n + 1) + [(n + 1) − j]

n+1 1 n 2

= 2 [2(n + 1) + (n − 2)(n + 3)] = 2 [n + 5n − 2], n ≥ 2

n−1 2

We have that : h(Z4 × Z2n−1 ) = 2 [(n − 1) + 5(n − 1) − 2]

n−1 2

= 2 [n + 3n − 6], n > 2 □

Corrolary 1 : Following the last proposition, h(Z4 × Z25 ), h(Z4 × Z26 ), h(Z4 × Z27 ) and h(Z4 × Z28 ) = 1536, 4096, 10496 and 26112 respectively.

Theorem A ( see [ 15 ] ) : Let G = D2n × C2, the nilpotent group formed by the cartesian product of the dihedral group of order 2 and a cyclic group of order 2. Then, the number of distinct

2n n+1

fuzzy subgroups of G is given by : h(G) = 2 (2n + 1) − 2 , n > 3

Proof:

The group D2n × C2, has one maximal subgroup which is isomorphic to Z2 × Z n−2 , two maximal subgroups which are isomorphic to D n−1 × C2, and 2 which are isomorphic to D2n .

It thus, follows from the Inclusion-Exclusion Principle using equation,

h(D2n × C2) = h(Z2 × Z n−1 ) + 4h(D2n ) − 8h(D n−1 ) − 2h(Z2 × Z n−2 ) + 2h(D n−1 × C2)

By recurrence relation principle we have :

2n n+1

h(D2n × C2) = 2 (2n + 1) − 2 , n > 3

By the fundermental principle of mathematical induction, set F(n) = h(D2n × C2), assuming the truth of F(k) =h(D k × C2) = 2h(Z2 × Zk−1)

2k k+1

+ 8h(D k − 16hD k−1 − 4h(Z2 × Zk−2) + 4h(D k−1 × C2) = 2 (2k + 1) − 2 , F(k+1) = h(D k+1 × C2) = 2h(Z2 × Z k ) + 8h(D k+1 − 16h(D k − 4h(Z2 × Zk−1)

2 2k k

+ 4h(D k × C2) = 2 [2 (2k − 3) − 2 ], which is true. □

Proposition 2 ( see [ 12 ] ) : Suppose that G = D2n × C4. Then, the number of distinct fuzzy subgroups of G is given by :

2(n−2) (n−1+j)

2 (64n + 173) + 3 2 (2n + 1 − 2j)

h(D2n × C4) = h(D2n × C2) + 2h(D n−1 × C4) − 4h(D n−1 × C2) + h(Z4 × Z n−1 ) − 2h(Z2 × Z2n−1 ) − 2h(Z4 × Z2n−2 ) + 8h(Z2 × Z2n−2 ) + h(Z2n−1 ) − 4h(Z2n−2 )

2n+2 2(n−3) n n+1 n+2 2(n−2)

h(D2n × C4) = (n − 3).2 + 2 (1460) + 3[2 (2n − 1) + 2 (2n − 3) + 2 (2n − 5) + · · · + 7(2 )]

2n+2 2(n−3) n−1+j

= (n − 3).2 + 2 (1460) + 3 2 (2n + 1 − 2j)

2(n−2) n−1+j

= 2 (64n + 173) + 3 2 (2n + 1 − 2j)

Proposition 3 ( see [10] ) : Let G be an abelian p-group of type Zp × Zp × Zpn , where p is a prime and n ≥ 1. The number of distinct fuzzy subgroups of G is

n n 3 n+1 3 n 3 2

h(Zp × Zp × Zpn ) = 2 p(p + 1)(n − 1)(3 + np + 2p) + (2 − 2)p − 2 (n − 1)p + 2 [p + 4(1 + p + p )].

Proof: There exist exactly 1 + p + p maximal subgroups for the abelian type Zp × Zp × Zpn , [Berkovich(2008)]. One of them is isomorphic to Zp × Zp × Z n−1 , while each of the remaining p + p is isomorphic to Zp × Zpn . Thus, by the application of the Inclusion-Exclusion Principle,we have as follows: h(Zp × Zp × Zpn ) =

n n 3 n+1 3 n 3 2

2 p(p + 1)(n − 1)(3 + np + 2p) + (2 − 2)p − 2 (n − 1)p + 2 [p + 4(1 + p + p )] And thus,

n−2 2 2 2 3 2

h(Zp × Zp × Zpn−2 ) = 2 [4 + (3n − 5)p + (n − 5)p + (n − 5n + 8)p ] − 2p .

Corrolary 2 : From (3) above, obsreve that, we are going to have that:

n+1 2

h(Z3 × Z3 × Z3n ) = 2 [18n + 9n + 26] − 54

Similarly, for p = 5, using the same analogy, we have

h(Z5 × Z5 × Z5n ) = 2[30h(Z5 × Z5n ) + h(Z5 × Z5 × Z n−1 ) −p h(Z5n ) − 30h(Z n−1 ) + 125]

And for p = 7,

h(Z7 × Z7 × Z7n ) = 2[56h(Z7 × Z7n ) + h(Z7 × Z7 × Z n−1 ) − 343h(Z7n ) − 56h(Z n−1 ) + 343]

n−2 2 2 2 3 2

We have, in general, h(Zp × Zp × Zpn−2 ) = 2 [4 + (3n − 5)p + (n − 5)p + (n − 5n + 8)p ] − 2p □

Proposition ( see [14] ) :

Let G = (D 3 × C2m ) for m ≥ 3 . Then , h(G) = m(89 − 23m) + (85)2 − 124

There exist seven maximal subgroups , of which one is isomorphic to D 3 × C2m−1, two being isomorphic to C2m × C2 × C2), two isomorphic to C2m × C2, and one each isomorphic to C2m × C4, and C2m respectively.

Hence , by the inclusion - exclusion principle, using the propositions [1], [2], [3], and Theorem [1] we have that :

h(G) = h(D 3 × C m−1 ) + 2h(C2m × C2) × C2) + 2h(C2m × C2) + 2h(C2m × C4) + h(C2m ) − 12h(C2m × C2) − 6h(C m−1 × C2) × C2) − 3h(C m−1 ) × C4) + 28h(C m−1 × C2) + 2h(C m−1 × C2) × C2) + 4h(C2m × C2) + h(C2m−1 × C4) − 35h(C2m−1 × C2) − 7h(C2m−1 × C2) + h(C2m−1 × C2) = h(D 3 × C ) + 2h(C2m × C2) × C2) − 6h(C2m × C2) + h(C2m × C4) + h(C2m ) − 4h(C m−1 × C2) × C2) − 2h(C2m−1 ) × C4) + 8h(C2m−1 × C2)

m−1 m+2 2 m m m+2 2 6 m 2

= h(D23 × C ) + 2 (6m + 7m + 9) − 32 − (6)2 (2m + 2) + 8m(2 ) − 2 6m − 5m + 8 + 2 + 2 (m +

m 2 m m−1 m m m−1 m

5m−2)−2 (3m+m −6) + 2 = h(D23 ×C ) + 2 (46m−4) + 2 + 32 = h(D23 ×C ) + 2 (46m−3) + 32

m−1 m+1 m+1 m

Hence , h(G) = 2h(D23 × C ) + 2 (46m − 3) + 64 = 2 (46m − 3) + 64 + 2[2 (46m −

m−2 m+1 m+1 7 2 m−2

49) + 64 + 2h(D23 × C )] = 2 (46m − 3) + 64 + 2 (46m − 49) + 2 + 2 h(D23 × C )

m+1 6 m+1 7 2 m−1 m−3

= 2 (46mm − 3) + 2 + 2 (46m − 49) + 2 + 2 [2 (46m − 95) + 64 + 2h(D23 × C )

m+1 6 m+1 7 m+1 8 3

h(D23n × C2m ) = (46m − 3).2 + 2 + (46m − 49)2 + 2 + (46m − 95)2 + 2 + 2 h(D 3 × C m−3 )

m+1 6 7 8 3 m−3

= 2 .[(46m − 3) + (46m − 49) + (46m − 95)] + 2 + 2 + 2 + 2 h(D23 × C )

6 7 8 5+k

= 2 + 2 + 2 + · · · + 2 | {z } series (1) + 2 .[46mk+ (−3 − 49 − 95 · · · (−3 − 46(k − 1)))] | {z } series (2) +2 h(D 3 × C2m−k), k ∈ {1, 2.3. · · · n ∈ N} 6 m−1 5+k

For the series (1) , we have that, Um = 2 .2 = 2 , m + 5 = k + 5, ⇒ m = k. We have that

6 2 −1 6 k

Sm=k = 2 [ ] = 2 (2 .1)

And for the second series (2), we have that , Tm = −3 + (m − 1)(−46) = −3 − 46(k − 1) ⇒ m − 1 =

k k k

k − 1, n = k Hence , Sm = k = [2(−3) + (k − 1)(−46)] = (−6 − 46k + 46) = (40 − 46k), We have that

k 6 k k

h(D23n ×C2m ) = (40−46k) + 2 (2 .1) + 2 h(D3 ×C2m−k. By setting m = k we have that k = m−3.Hence

6 m−3 m

, h(D 3 × C2m ) = (m − 3)(20 − 23m) + 2 (2 − 1) + 2 − 3h(D3 × C 3 )

6 m−3 m−3 m−3 6 m+5

h(G) = (m − 3)(20 − 23m) + 2 (2 − 1) + 2 (5376) = (m − 3)(20 − 23m) + 2 − 2 + 2 (21)

2 m+3 6 m+5 2 m+3 6 m+5

= 20m − 23m − 60 + 69m + 2 − 2 + (21)2 = (89m − 23m − 60) + 2 − 2 + (21)2 = m(89 − 23m) − 124 + (85)2 □

1 n+1 3 2

Theorem ( see [11] ) : Let G = Z2n × Z8, then h(G) = (2 )(n + 12n + 17n − 24) Proof : The three maximal subgroups of G have the following properties :

one is isomorphic to Z8 × Z n−1 ), while two are isomorphic to Z4 × Z2n ) .

We have : h(G) = 2h(Z4 × Z2n ) + h(Z8 × Z n−1 ) − 3h(Z4 × Z n−1 ) + h(Z4 × Z n−1 ) = 2h(Z4 × Z2n ) + h(Z8 × Z n−1 ) − 2h(Z4 × Z n−1 ) = h(Z8 × Z n−1 ) + 2h(Z4 × Z2n ) − h(Z4 × Z n−1 ) Hence , h(G) = 4h(Z4 × Z2n ) − 4h(Z4 × Z n−1 ) + 2h(Z8 × Z n−1 ) = 4h(Z4 × Z2n ) + 4h(Z4 × Z n−1 ) + 8h(Z4 × Z n−2 ) − 16h(Z4 × Z n−3 ) + 32h(Z4 × Z2n−3 ) − 32h(Z4 × Z2n−4 ) + 16h(Z8 × Z2n−4 ) = 4h(Z4 × Z2n ) + 4h(Z4 × Z n−1 ) + 8h(Z4 × Z n−2 ) + 16h(Z4 × Z n−3 ) + 32h(Z4 × Z2n−4 ) − 64h(Z4 × Z2n−5 ) + 32h(Z8 × Z2n−5 ) + · · · − 2 h(Z4 × Z2n−j ) + 2 h(Z8 × Z2n−j ) , for n − j = 3

n−3 n−1 k+1

= 4h(Z4 × Z2n ) + 2 h(Z8 × Z 3 ) − 2 h(Z4 × Z 3 ) + [2 h(Z4 × Z n−k )

Pn−3

n+2 2 n+2 2 1 2

= 2 [n + 5n + 3] + h(Z4 × Z n−k ) = 2 ((n + 5n + 3) + (n − 3)(n + 9n + 14))

k=1 2 6 1 n+1 3 2

= (2 )(n + 12n + 17n − 24), n > 2. □

Proposition ( see [16] : Suppose that G = D2n × C8. Then, the number of distinct fuzzy subgroups of G is given by :

2(n−1) n 2 (j−1j)

2 (6n + 113) + 2 [13 − 6n − 2n + 3 2 (2n + 1 − 2j)]

n+2 3 3 2 k 3 2

+ (2 )[(n − 1) + (n − 2) + 24n − 38n − 30 + 2 [(n − 2 − k) + 12(n − 2 − k) + 17(n − k) − 58]]

Proof : h(D2n × C8) = 2h(Z n−1 ) + 2h(D2n × Z4) + 2h(D n−1 × C8) + 4h(Z n−2 × C8) + 2 h(Z n−3 × C8) + 2 h(Z n−4 × C8) − 2 h(Z n−5 × Z23 ) − 4h(Z n−1 × Z22 ) + 2 h(Z n−5 ) × Z22 − 2 h(Z n−5 ) − 2 h(D n−4 × C22 ) + 2 h(D n−4 × C23 ) = 2 + 2h(D2n × C4) + 2h(Z n−1 × Z 3 ) + 2 h(Z n−2 × Z 3 )

2(n−3) 2(n−2) 2 2n−5

− 2 h(Z22 × Z23 ) + 2 h(Z22 × Z22 − 2 h(Z n−1 × Z22 ) − 2 h(Z22 )

2n−5 2(n−3)

− 2 h(D23 × Z22 ) + 2 h(D23 × Z23 )

2ij

+ 3 2 h(Z n−2−i × Z23 ) as required. □

Theorem : Let G = D24 × C24 . Then , h(G) = 61384

Proof : There exist seven maximal subgroups . Two isomorphic to D24 × C23 . two isomorphic to D23 × C24 . two isomorphic to D24 × C22 , while the seventh is isomorphic to Z24 .

Hence , we have that : h(G) = 2h(D24 × Z22 ) + 2h(D24 × Z23 ) + 2h(D23 × Z24 ) − 6h(D23 × Z23 ) − 6h(Z24 × Z22 ) − 3h(Z23 × Z23 ) − 6h(Z24 ) + 2h(D23 × Z23 ) + 28h(Z23 × Z22 ) + 2h(Z24 × Z22 ) + 2h(Z24 ) + h(Z23 × Z23 ) − 35h(Z23 × Z22 ) + 21h(Z23 × Z22 ) − 7h(Z23 Z22 ) + h(Z23 × Z22 ) = 2[h(D24 × Z22 ) + h(D24 × Z23 ) + h(D23 × Z24 ) − 2h(D23 × Z23 ) − 2h(Z24 × Z22 ) − h(Z23 × Z23 ) + 4h(D23 × Z22 ) − 3h(Z24 ) + h(Z24 )]

∴ h(G) = 4[700 + 8416 + 10744 − 10752˘1088 + 162 + 704˘40] = 4[15346] = 61384 □

3|Computation for G = D 4 × C n, n ≥ 4.

Our computation on the algebraic fuzzy structure given actually has an outcome which involves multiple sums

The maximal subgroups are :

(D 4 × C n−1 ), 2(D 3 × C2n ), 2(D2n × C 2 ), (D2n × C 3 ) and (C2n ).

2 2 2 2 2

We have that : h(G) = h(D 4 × Cn−1) + 2h(D 3 × Cn) + 2h(D2n × C 2 ) + h(D2n × C 3 ) + h(C2n ) −

2 2 2 2 2

6h(D 3 × Z n−1 ) − 6h(Z2n × Z 2 ) − 3h(Z n˘1 × Z 3 ) − 6h(Z2n ) + 2h(D 3 × C n−1 ) + 28h(C n−1 × C2n ) +

2 2 2 2 2 2 2 2

h(C n−1 ×C 3 )+2h(C2n ×C 2 )+2h(Z2n )−35h(C n−1 ×C 2 )+21h(C n−1 ×C 2 )−7h(C n−1 ×C 2 )+h(C n−1 ×C 2 )

2 2 2 2 2 2 2 2 2 2 2

= h(D 4 × C n−1 ) + 2h(D 3 × C ) + 2h(D2n × C 2 ) + h(D2n × C 3 ) − 4h(D 3 × Z n−1 ) − 4h(Z2n × Z 2 ) −

2 2 2 2 2 2 2 2 2

2h(Z n˘1 × Z 3 ) + 8h(Z n−1 × Z 2 ) − 3h(Z2n )

h(G) = h(D 4 × Z n−k ) + 2h(D 3 × Z2n ) − 4h(D 3 × Z n−k ) − 4h(Z2n × Z 2 )

2 2 2 2 2 2

k k k

X X X −2h(Z n−k × Z23 ) + 8h(Z n−k × Z22 ) + h(D n−1+j × Z23 ) + 2 h(D n−1+j × Z22 ) − 3 h(Z n+1−j )

2 2 2 2 2 j=1 j=1 j=1

k−1 k−1 k−1

X X X −2 h(D 3 × Z n−j ) + 4 h(D n−j × Z 2 ) − 2 h(D2n−j × Z 3 ),

2 2 2 2 2 j=1 j=1 j=1

whence , n − k = 4, ⇒ k = n − 4. ∴ h(G) = 2h(D 4 × Z 4 ) + 4h(D 3 × Z2n ) − 8h(D 3 × Z 4 ) − 8h(Z2n × Z2n ) −

2 2 2 2 2

4h(Z24 × Z23 ) + 16h(Z24 × Z22 )+

n−4 n−4 n−4

X X X 2 h(D n−1+j × Z23 ) + 4 h(D n−1+j × Z22 ) − 6 h(Z n+1−j )

j=1 j=1 j=1 n−5 n−5 n−5

X X X −4 h(D 3 × Z n−j ) + 8 h(D2n−j × Z 2 ) − 4 h(D n−j × Z 3 )

2 2 2 2 2 j=1 j=1 j=1 n+3 2 2

∴ h(G) = 2 (422 − n − 5n) − 9n + 356n − 29160 + 2 h(D n−1+j × Z23 )

n−4 n−4 n−5 n−5 n−5

X X X X X +4 h(D n−1+j ×Z 2 )−6 h(Z n+1−j )−4 h(D 3 ×Z n−j ) + 8 h(D2n−j ×Z 2 )−4 h(D n−j ×Z 3 )

2 2 2 2 2 2 2 2 j=1 j=1 j=1 j=1 j=1

n+3 2 2

= 2 (422 − n − 5n) − 9n + 356n − 29160 + [2h(D n−1+j × Z23 ) + 4h(D n−1+j × Z22 ) − 6h(Z n+1−j )]

− [4h(D 3 × Z n−j ) − 8h(D2n−j × Z 2 ) + 4h(D n−j × Z 3 )]

2 2 2 2 2

Hence , proved as required □

The computations so far by the use of GAP ( General AlgorithmAlgorithms and Programming ) and the Inclusion - Exclusion Principle can be certified here as being very useful in the computations of the distinct number of fuzzy subgroups for the finite nilpotent p - groups .

We have the following examples as parts surfacing from our computations so far. The readers may consider the examples below in tabular format.

Example 1 :

Table 1 Table Summarizing some Number of Distinct Fuzzy Subgroups of (D 3 × C2n ) FOR ≥ 3

S/N for the Number of m 3 4 5 6 7 8 9 10 h(G) = (D 3 × C2n ), n ≥ 3 5376 10728 21506 43347 86536 173320 347098 694910

Example 2 : Now, since the stipulated condition that m ≥ 3 must definitely be fulfilled then the readers may consider the examples below in tabular format.

Table 2 Table Summarizing some Number of Distinct Fuzzy Subgroups of (D 4 × C2n ) FOR n ≥ 4

S/N for the Number of n 4 5 6 h(G) = (D 4 × C2n ), n ≥ 4 20, 200 375, 648 3, 893, 800

The discoveries from our studies so far , has helped to observe that any finite product of nilpotent group is nilpotent. Also, the problem of classifying the fuzzy subgroups of a finite group has experienced a very rapid progress. Finally, the method can be used in further computations up to the generalizations of similar and other given structures

Funding: This research received no external funding

Competing of interests statement : The authors declare that in this paper, there is no competing of interests.

Acknowledgments The authors are grateful to the anonymous reviewers for their helpful comments and suggestions which has improved the overall quality and the earlier version of the work.