Classifying fuzzy (normal) subgroups of the group D 2 p × ℤ q and finite groups of order n ≤ 20

Abstract

One of the problems in fuzzy group theory is concerned with classifying the fuzzy (normal) subgroups of a finite group. In this paper, we classify the fuzzy (normal) subgroups of $D_{2 p} \times ℤ_{q}$ (p, q primes) and all finite groups of order n ≤ 20, by using the natural equivalence relation. In this case, the corresponding equivalence classes of fuzzy (normal) subgroups of a group G are closely connected to the chains of (normal) subgroups in G. In fact, for determining the number of these classes, we calculate the number of all chains of (normal) subgroups of G that terminate in G.

Keywords

Fuzzy subgroup chain of subgroups level subgroups dihedral group dicyclic group

1 Introduction

Classifying the fuzzy subgroups and fuzzy normal subgroups of a finite group has undergone a rapid development, in the last years. Many papers have treated the particular cases of finite groups. In [8], Murali and Makamba studied equivalence classes of fuzzy subgroups of a given group under a suitable equivalence relation. They characterized the number of fuzzy subgroups of finite abelian groups. In [9], the number of distinct fuzzy subgroups of a finite cyclic group of square-free order is determined, while [10 –12] and [20] deal with this number for cyclic groups of order pⁿq^m (p,q primes). Recall here the paper [15], where 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: finite cyclic groups and finite elementary abelian p-groups. The explicit formula obtained for the first class leads in to an expression of the well-known central Delannoy numbers [14]. Also, the above study has been extended in to some remarkable classes of non-abelian groups: dihedral groups, symmetric groups, finite p-groups having a cyclic maximal subgroup, hamiltonian groups, dicyclic groups and non-abelian groups of order p³ and 2⁴ in [16–18 , 3] and [2, 4], respectively.

After these, T $\overset{˘}{a}$ rn $\overset{˘}{a}$ uceanu in [19] deals with classifying the fuzzy normal subgroups of symmetric groups and dihedral groups. In [5], this number is calculated for the particular cases of non-abelian finite groups.

Without any equivalence on fuzzy subsets of a set, the number of fuzzy subgroups of a finite group is infinite even this number for the trivial group {e} is infinite. In the present paper, we use the natural equivalence relation introduced in [20]. We determine the number of distinct fuzzy (normal) subgroups of $D_{2 p} \times ℤ_{q}$ (p, q primes) and all finite groups of order n ≤ 20 with respect to this equivalence. In this case, it is sufficient to find the number of all chains of (normal) subgroups of G that terminate in G. In many situations this leads us to some recurrence relations, whose solutions can be easily found. Note that an essential role in counting of fuzzy subgroups will be played by the Inclusion-Exclusion Principle. As a guiding principle in determining the number of these classes, we first find the number of maximal chains of subgroups of G.

This paper is organized as follows. In Section 2, we recall some preliminary definitions and present the main results of fuzzy (normal) subgroups of [1 , 19]. In Section 3, we calculate the numbers of maximal chains of subgroups and distinct fuzzy (normal) subgroups of $D_{2 p} \times ℤ_{q}$ . Section 4 deals with characterizing the number of distinct fuzzy subgroups and distinct fuzzy normal subgroups of all finite groups of order n ≤ 20.

2 Preliminaries

In this section, we present some basic notions and results of fuzzy (normal) subgroup theory. Let us first recall preliminary definitions (for more details, see [6 , 13]).

Definition 2.1. Let G be a group and μ be a fuzzy subset of G. Then, μ is called a fuzzy subgroup of G if it satisfies the following conditions:

min {μ (x) , μ (y)} ≤ μ (xy) , for all x, y∈ G ;

μ (x) ≤ μ (x^-1) , for all x ∈ G .

It is easy to see that μ (x^-1) = μ (x) and μ (e) ≥ μ (x), for all x ∈ G, where e is the identity element of G. A fuzzy subgroup μ is called normal, if μ (xy) = μ (yx), for all x, y ∈ G.

If μ is a fuzzy subgroup of G, then the following conditions are equivalent, for all x, y ∈ G:

μ is normal,

μ (xyx^-1) ≥ μ (y) ,

μ (xyx^-1) = μ (y) .

The set FL (G) (FN (G), respectively) consisting of all fuzzy subgroups (fuzzy normal subgroups, respectively) of G forms lattice with respect to fuzzy set inclusion.

For each t ∈ [0, 1], we define the level subset: $U (μ, t) = {x \in G μ (x) \geq t} .$

These subsets allow us to characterize the fuzzy (normal) subgroups of G, in the following manner: Let G be a group and μ be a fuzzy subset of G. Then, μ is a fuzzy (normal) subgroup of G if and only if its non-empty level subsets are (normal) subgroups of G. The fuzzy (normal) subgroups of G can be classified up to some natural equivalence relations on the set consisting of all fuzzy subsets of G. The following is one of them:

Let μ and η be two fuzzy subsets of G. Then, we define $\begin{matrix} μ \sim η if and only if (μ (x) > μ (y) \Leftrightarrow \\ η (x) > η (y), for all x, y \in G); \end{matrix}$ and two fuzzy (normal) subgroups μ and η of G will be called distinct if μnotsimη.

Let G be a finite group and μ be a fuzzy subgroup of G. Suppose that Imμ = {t₁, t₂, …, t_r} such that t₁ > t₂ > … > t_r. Then, μ determines the following chain of subgroups of G which ends in G: $\begin{matrix} U (μ, t_{1}) \subset U (μ, t_{2}) \subset \dots \subset U (μ, t_{r}) = G . \end{matrix}$

Moreover, for any x ∈ G and i = 1, …, r we have $\begin{matrix} μ (x) = t_{i} & \Leftrightarrow & i = max {j | x \in U (μ, t_{j})} \\ \Leftrightarrow & x \in U (μ, t_{i}) ∖ U (μ, t_{i - 1}), \end{matrix}$ where by convention, we set U (μ, t₀) =∅. Let G be a group and μ, η be two fuzzy (normal) subgroups of G. Then, a necessary and sufficient condition for fuzzy normal subgroups μ, η of G to be equivalent with respect to ∼ is that they determine the same chain of (normal) subgroups of G which ends in G. This result shows that there exists a bijection between the equivalence classes of fuzzy (normal) subgroups of G and the set of chains of (normal) subgroups of G which end in G. So, there is a link between FL (G) (FN (G), respectively) and the classical subgroup lattice L (G) (normal subgroup lattice N (G), respectively) of G. In fact, the problem of counting all distinct fuzzy (normal) subgroups of G can be translated in to a combinatorial problem: finding the number of all chains of (normal) subgroups of G that terminate in G. Throughout this paper, F (G) (F_N (G), respectively) denotes the number of all distinct fuzzy subgroups (fuzzy normal subgroups, respectively) of the group G.

The largest class of groups for which it was completely solved is constituted by finite cyclic groups.

Theorem 2.2. [15, Corollary 4] If G is a finite cyclic group of order n (that is $G ≅ ℤ_{n}$ ) and $n = p_{1}^{m_{1}} p_{2}^{m_{2}} \dots p_{s}^{m_{s}}$ is the decomposition of n as a product of prime factors, then the number of all distinct fuzzy subgroups of G is given by the equality: $F (ℤ_{n}) = 2^{\sum_{α = 1}^{s} m_{α}} \sum_{i_{2} = 0}^{m_{2}} \sum_{i_{3} = 0}^{m_{3}} \dots \sum_{i_{s} = 0}^{m_{s}} {(- 1 / 2)}^{\sum_{α = 2}^{s} i_{α}} K_{s},$ where $K_{s} = \prod_{α = 2}^{s} (\begin{matrix} m_{α} \\ i_{α} \end{matrix}) (\begin{matrix} m_{1} + \sum_{β = 2}^{α} (m_{β} - i_{β}) \\ m_{β} \end{matrix})$ and the above iterated sums are equal to 1 for s = 1.

Corollary 2.3. [15, Remark 6] The number of all distinct fuzzy subgroups of the finite cyclic group G of order pⁿq^m (p, q primes) is given by the equality: $F (G) = 2^{n + m} \sum_{r = 0}^{m} (1 / 2^{r}) (\begin{matrix} n \\ r \end{matrix}) (\begin{matrix} m \\ r \end{matrix}) .$

Especially, the number of all distinct fuzzy subgroups of the finite cyclic p-group G (i.e. when m = 0) is 2ⁿ.

The problem of counting number of all distinct fuzzy subgroups is solved for several groups. In the following, we review some results which we will need in the next sections.

Theorem 2.4. [1, Section 2] The number of all distinct fuzzy subgroups of the group $ℤ_{2} \times ℤ_{2} \times ℤ_{p^{n}}$ is given by the equality: $\begin{matrix} F (ℤ_{2} \times ℤ_{2} \times ℤ_{p^{n}}) \\ = 2^{n - 1} (n^{3} + 15 n^{2} + 56 n + 48) / 3 . \end{matrix}$

Let, $G = ℤ_{p}^{k}$ and H₁, H₂, …, H_{n
_k,1} be minimal subgroups of G, where $n_{k, 1} = \sum_{i = 1}^{k} p^{i - 1} = (p^{k} - 1) / (p - 1)$ . Then, for all r ∈ {1, 2, …, n_k,1} and 1 ≤ s ≤ min {r, k}, the cardinality of the set $\begin{matrix} {(i_{1}, i_{2}, \dots, i_{r}) | 1 \leq i_{1} < i_{2} < \dots < i_{r} \leq n_{k, 1}; \\ | H_{i_{1}} + H_{i_{2}} + \dots + H_{i_{r}} | = p^{s}} . \end{matrix}$ is denoted by x_r,s (see [15, Proposition 8], for more details).

Theorem 2.5. [15, Theorem 9] The number of all distinct fuzzy subgroups of the group $ℤ_{p}^{k}$ is given by the equality: $\begin{matrix} F (ℤ_{p}^{k}) = 2 \sum_{r = 1}^{n_{k, 1}} (- 1)^{r - 1} \sum_{s = 1}^{min {r, k}} x_{r, s} F (ℤ_{p}^{k - s}) . \end{matrix}$

Especially, $F (ℤ_{p}^{2}) = 2 p + 4$ and $F (ℤ_{p}^{3}) = 2 p^{3} + 8 p^{2} + 8 p + 8$ .

The following theorem is used for counting the number of all chains of subgroups of G that terminate in G.

Theorem 2.6. [16, Section 2] Let G be a group and M₁, M₂, …, M_k be maximal subgroups of G. Then, F (G) is given by the following equality: $\begin{matrix} F (G) & = & 2 (\sum_{i = 1}^{k} F (M_{i}) - \sum_{1 \leq i_{1} < i_{2} \leq k} F (M_{i_{1}} \cap M_{i_{2}}) \\ + \dots + (- 1)^{k - 1} F (⋂_{i = 1}^{k} M_{i})) . \end{matrix}$

If maximal subgroup structure of G is known, in some cases, Theorem 2 will lead to recurrence relations that permit us to determine F (G). The above theorem will play an essential role in the next theorems to solve counting problem for finite dihedral groups and dicyclic groups.

Remind that the dihedral group D_2n is the symmetry group of a regular polygon with n sides and has the order 2n. The most convenient abstract description of D_2n is obtained by using its generators: a rotation x of order n and a reflection y of order 2. Under these notations, we have D_2n = 〈x, y | xⁿ = y² = e, y^-1xy = x^-1〉. The dicyclic group T_4n (n > 1) is a subgroup of the unit quaternions generated by x = exp (iπ/n) = cos (π/n) + isin (π/n), y = j. More abstractly, one can define the dicyclic group T_4n as any group having the presentation T_4n = 〈x, y | x²ⁿ = e, y² = xⁿ, y^-1xy = x^-1〉. The generalized quaternion group $\begin{matrix} Q_{2^{a + 1}} \\ = 〈 x, y | x^{2^{a}} = e, y^{2} = x^{2^{a - 1}}, y^{- 1} xy = x^{- 1} 〉, \end{matrix}$ where a > 1, is a subclass of dicyclic group.

Theorem 2.7. (1) The number of all distinct fuzzy subgroups of the dihedral group

[16, Theorem 4.3] $D_{2 p_{1}^{m}}$ is given by the equality $F (D_{2 p_{1}^{m}}) = 2^{m} (p_{1}^{m + 1} + p_{1} - 2) / (p_{1} - 1)$ . Especially, F (D_2^m+1) =2^2m+1.

[16, Theorem 4.4] D_2p₁p₂ is given bythe equality F (D_2p₁p₂) =2 (3p₁p₂ + 2p₁ + 2p₂ + 6).

(2) [19, Theorem 4] The number of all distinct fuzzy normal subgroups of the dihedral group D_2n, n ≥ 3, is given by the equality: $\begin{matrix} F_{N} (D_{2 n}) \\ = {\begin{matrix} 1 + \sum_{d | n} F (ℤ_{d}), & n \equiv 1 (\mod 2) \\ 3 + \sum_{d | n} F (ℤ_{d}) + 2 \sum_{d | (n / 2)} F (ℤ_{d}), & n \equiv 0 (\mod 2) . \end{matrix} \end{matrix}$

Especially, we have F_N (D_2^k+1) =2^k+2.

Theorem 2.8. (1) [3, Theorem 4.1] The number of all distinct fuzzy subgroups of the dicyclic group T_4p^m is given by the equality: $\begin{matrix} F (T_{4 p^{m}}) \\ = {\begin{matrix} 4^{m + 1}, & p = 2 \\ 2^{m} (m + 2 + \sum_{i = 0}^{m} (m + 2 - i) p^{i}), & p \neq 2 . \end{matrix} \end{matrix}$

(2) [5, Theorem 3.2] The number of all distinct fuzzy normal subgroups of the dicyclic group T_4n, n ≥ 2, is given by the equality: $\begin{matrix} F_{N} (T_{4 n}) \\ = {\begin{matrix} 1 + \sum_{d | 2 n} F (ℤ_{d}), & n \equiv 1 (\mod 2) \\ 3 + \sum_{d | 2 n} F (ℤ_{d}) + 2 \sum_{d | n} F (ℤ_{d}), & n \equiv 0 (\mod 2) . \end{matrix} \end{matrix}$

Especially, we have F_N (T_2^k+2) =2^k+3.

3 The number of fuzzy (normal) subgroups of finite group $D_{2 p} \times ℤ_{q}$

In this section, we give an explicit formula for the number of fuzzy (normal) subgroups of the finite group $D_{2 p} \times ℤ_{q}$ . For this purpose, we consider three cases.

Case 1: q = 2.

Consider the group

G = D_{2 p} \times ℤ_{2}

i.e.,

\begin{matrix} 〈 x, y | x^{p} = e, y^{2} = e, y^{- 1} xy = x^{- 1} 〉 \times {0, 1}, \end{matrix}

where p is an odd prime number. According to the representation of G, we get

Order	Number
2	2p + 1
p	p - 1
2p	p - 1

G has p + 3 maximal subgroups such that they are of prime index 2 or p. These are as follows: $\begin{matrix} M_{1} = 〈 (x, 1) 〉 ≅ ℤ_{2 p}; \\ M_{2} = 〈 (x, 0); (y, 0) 〉 ≅ D_{2 p}; \\ M_{3} = 〈 (x, 0); (y, 1) 〉 ≅ D_{2 p}; \\ H_{1} = 〈 (e, 1); (y, 1) 〉 ≅ ℤ_{2} \times ℤ_{2}; \\ H_{2} = 〈 (e, 1); (xy, 1) 〉 ≅ ℤ_{2} \times ℤ_{2}; \\ ⋮ \\ H_{p} = 〈 (e, 1); (x^{p - 1} y, 1) 〉 ≅ ℤ_{2} \times ℤ_{2} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(e, 0)} \subseteq 〈 (e, 1) 〉 \subseteq M_{1} \subseteq G; \\ {(e, 0)} \subseteq 〈 (x, 0) 〉 \subseteq M_{i} \subseteq G, 1 \leq i \leq 3; \\ {(e, 0)} \subseteq 〈 (x^{i} y, 0) 〉 \subseteq M_{2} \subseteq G, 0 \leq i \leq p - 1; \\ {(e, 0)} \subseteq 〈 (x^{i} y, 1) 〉 \subseteq M_{3} \subseteq G, 0 \leq i \leq p - 1; \\ {(e, 0)} \subseteq 〈 (e, 1) 〉 \subseteq H_{i} \subseteq G, 1 \leq i \leq p; \\ {(e, 0)} \subseteq 〈 (x^{i - 1} y, j) 〉 \subseteq H_{i} \subseteq G, 1 \leq i \leq p, \\ 0 \leq j \leq 1 . \end{matrix}$

Then, we have $\begin{matrix} M_{1} \cap M_{2} = M_{1} \cap M_{3} = M_{2} \cap M_{3} \\ = 〈 (x, 0) 〉 ≅ ℤ_{p}; \\ M_{1} \cap H_{i} = 〈 (e, 1) 〉 ≅ ℤ_{2}, 1 \leq i \leq p; \\ M_{2} \cap H_{i} = 〈 (x^{i - 1} y, 0) 〉 ≅ ℤ_{2}, 1 \leq i \leq p; \\ M_{3} \cap H_{i} = 〈 (x^{i - 1} y, 1) 〉 ≅ ℤ_{2}, 1 \leq i \leq p; \\ H_{i} \cap H_{j} = 〈 (e, 1) 〉 ≅ ℤ_{2}, 1 \leq i, j \leq p and i \neq j; \\ M_{1} \cap M_{2} \cap M_{3} = 〈 (x, 0) 〉 ≅ ℤ_{p}; \\ M_{1} \cap H_{i} \cap H_{j} = 〈 (e, 1) 〉 ≅ ℤ_{2}, 1 \leq i, j \leq p, i \neq j; \\ H_{i} \cap H_{j} \cap H_{k} = 〈 (e, 1) 〉 ≅ ℤ_{2}, 1 \leq i, j, k \leq p, \\ i \neq j \neq k . \end{matrix}$

For 4 ≤ i ≤ p + 1, we find that $\begin{matrix} M_{1} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 1}} = 〈 (e, 1) 〉 ≅ ℤ_{2}, \\ H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i}} = 〈 (e, 1) 〉 ≅ ℤ_{2}, \end{matrix}$ where 1 ≤ j₁, j₂, …, j_i-1, j_i ≤ p and j₁ ≠ j₂ ≠ … ≠ j_i-1 ≠ j_i. Other intersections of the maximal subgroups are trivial group 〈 (e, 0) 〉. Therefore, by Theorem 2.6, we obtain $\begin{matrix} (1 / 2) F (G) \\ = F (ℤ_{2 p}) + 2 F (D_{2 p}) + pF (ℤ_{2} \times ℤ_{2}) \\ - 2 F (ℤ_{p}) + ((\begin{matrix} p \\ 3 \end{matrix}) - 3 p) F (ℤ_{2}) \\ + \sum_{i = 4}^{p + 1} (- 1)^{i - 1} (\begin{matrix} p + 1 \\ i \end{matrix}) F (ℤ_{2}) \\ + \sum_{i = 4}^{p + 1} (- 1)^{i - 1} ((\begin{matrix} p + 3 \\ i \end{matrix}) - (\begin{matrix} p + 1 \\ i \end{matrix})) \\ + (p + 2) (p + 1) . \end{matrix}$

Thus, by Corollary 2.3 and Theorems 2.5, 2.7, F (G) =16p + 20.

Case 2: q = p.

Consider the group

G = D_{2 p} \times ℤ_{p}

i.e.,

\begin{matrix} 〈 x, y | x^{p} = e, y^{2} = e, y^{- 1} xy = x^{- 1} 〉 \\ \times {0, 1, \dots, p - 1}, \end{matrix}

where p is an odd prime number. According to the representation of G, we get

Order	Number
2	p
p	(p + 1) (p - 1)
2p	p (p - 1)

G has p + 2 maximal subgroups such that they are of prime index 2 or p. They are as follows: $\begin{matrix} M_{1} = 〈 (x, 0); (x, 1) 〉 ≅ ℤ_{p} \times ℤ_{p}; \\ M_{2} = 〈 (x, 0); (y, 0) 〉 ≅ D_{2 p}; \\ H_{1} = 〈 (y, 1) 〉 ≅ ℤ_{2 p}; \\ H_{2} = 〈 (xy, 1) 〉 ≅ ℤ_{2 p}; \\ ⋮ \\ H_{p} = 〈 (x^{p - 1} y, 1) 〉 ≅ ℤ_{2 p} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(e, 0)} \subseteq 〈 (e, 1) 〉 \subseteq M_{1} \subseteq G; \\ {(e, 0)} \subseteq 〈 (x, 0) 〉 \subseteq M_{2} \subseteq G; \\ {(e, 0)} \subseteq 〈 (x, i) 〉 \subseteq M_{1} \subseteq G, 0 \leq i \leq p - 1; \\ {(e, 0)} \subseteq 〈 (x^{i} y, 0) 〉 \subseteq M_{2} \subseteq G, 0 \leq i \leq p - 1; \\ {(e, 0)} \subseteq 〈 (e, 1) 〉 \subseteq H_{i} \subseteq G, 1 \leq i \leq p; \\ {(e, 0)} \subseteq 〈 (x^{i - 1} y, 0) 〉 \subseteq H_{i} \subseteq G, 1 \leq i \leq p . \end{matrix}$

We have $\begin{matrix} M_{1} \cap M_{2} = 〈 (x, 0) 〉 ≅ ℤ_{p}; \\ M_{1} \cap H_{i} = 〈 (e, 1) 〉 ≅ ℤ_{p}, 1 \leq i \leq p; \\ M_{2} \cap H_{i} = 〈 (x^{i - 1} y, 0) 〉 ≅ ℤ_{2}, 1 \leq i \leq p; \\ H_{i} \cap H_{j} = 〈 (e, 1) 〉 ≅ ℤ_{p}, 1 \leq i, j \leq p and i \neq j . \end{matrix}$

For 3 ≤ i ≤ p, we find that $\begin{matrix} M_{1} \cap M_{2} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 2}} = 〈 (e, 0) 〉, \\ M_{1} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 1}} = 〈 (e, 1) 〉 ≅ ℤ_{p}, \\ M_{2} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 1}} = 〈 (e, 0) 〉, \\ H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i}} = 〈 (e, 1) 〉 ≅ ℤ_{p}, \end{matrix}$ where 1 ≤ j₁, j₂, …, j_i-2, j_i-1, j_i ≤ p and j₁ ≠ j₂ ≠ … ≠ j_i-2 ≠ j_i-1 ≠ j_i. Therefore, by Theorem 2.6, we obtain $\begin{matrix} (1 / 2) F (G) \\ = F (ℤ_{p} \times ℤ_{p}) + F (D_{2 p}) + pF (ℤ_{2 p}) - p \\ - pF (ℤ_{2}) - (p + 2 + (\begin{matrix} p \\ 2 \end{matrix})) F (ℤ_{p}) \\ + \sum_{i = 3}^{p} (- 1)^{i - 1} (\begin{matrix} p + 1 \\ i - 1 \end{matrix}) \\ + \sum_{i = 3}^{p} (- 1)^{i - 1} (\begin{matrix} p + 1 \\ i \end{matrix}) F (ℤ_{p}) . \end{matrix}$

Thus, by Corollary 2.3 and Theorems 2.5, 2.7, F (G) =14p + 12.

Case 3: q ≠ 2, q ≠ p.

Consider the group

G = D_{2 p} \times ℤ_{q}

i.e.,

\begin{array}{l} 〈 x, y | x^{p} = e, y^{2} = e, y^{- 1} x y = x^{- 1} 〉 \\ \times {0, 1, \dots, q - 1}, \end{array}

where p, q are odd prime numbers and p ≠ q. According to the representation of G, we get

Order	Number
2	p
p	p - 1
q	q - 1
2q	p (q - 1)
pq	(p - 1) (q - 1)

G has p + 2 maximal subgroups such that they are of prime index 2, p or q. They are as follows: $\begin{matrix} M_{1} = 〈 (x, 1) 〉 ≅ ℤ_{pq}; \\ M_{2} = 〈 (x, 0); (y, 0) 〉 ≅ D_{2 p}; \\ H_{1} = 〈 (y, 1) 〉 ≅ ℤ_{2 q}; \\ H_{2} = 〈 (xy, 1) 〉 ≅ ℤ_{2 q}; \\ ⋮ \\ H_{p} = 〈 (x^{p - 1} y, 1) 〉 ≅ ℤ_{2 q} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(e, 0)} \subseteq 〈 (e, 1) 〉 \subseteq M_{1} \subseteq G; \\ {(e, 0)} \subseteq 〈 (x, 0) 〉 \subseteq M_{i} \subseteq G, 1 \leq i \leq 2; \\ {(e, 0)} \subseteq 〈 (x^{i} y, 0) 〉 \subseteq M_{2} \subseteq G, 0 \leq i \leq p - 1; \\ {(e, 0)} \subseteq 〈 (e, 1) 〉 \subseteq H_{i} \subseteq G, 1 \leq i \leq p; \\ {(e, 0)} \subseteq 〈 (x^{i - 1} y, 0) 〉 \subseteq H_{i} \subseteq G, 1 \leq i \leq p . \end{matrix}$

We have $\begin{matrix} M_{1} \cap M_{2} = 〈 (x, 0) 〉 ≅ ℤ_{p}; \\ M_{1} \cap H_{i} = 〈 (e, 1) 〉 ≅ ℤ_{q}, 1 \leq i \leq p; \\ M_{2} \cap H_{i} = 〈 (x^{i - 1} y, 0) 〉 ≅ ℤ_{2}, 1 \leq i \leq p; \\ H_{i} \cap H_{j} = 〈 (e, 1) 〉 ≅ ℤ_{q}, 1 \leq i, j \leq p and i \neq j . \end{matrix}$

For 3 ≤ i ≤ p, we find that $\begin{matrix} M_{1} \cap M_{2} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 2}} = 〈 (e, 0) 〉, \\ M_{1} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 1}} = 〈 (e, 1) 〉 ≅ ℤ_{q}, \\ M_{2} \cap H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i - 1}} = 〈 (e, 0) 〉, \\ H_{j_{1}} \cap H_{j_{2}} \cap \dots \cap H_{j_{i}} = 〈 (e, 1) 〉 ≅ ℤ_{q}, \end{matrix}$ where 1 ≤ j₁, j₂, …, j_i-2, j_i-1, j_i ≤ p and j₁ ≠ j₂ ≠ … ≠ j_i-2 ≠ j_i-1 ≠ j_i. Therefore, by Theorem 2.6, we obtain $\begin{matrix} (1 / 2) F (G) & = & F (ℤ_{pq}) + F (D_{2 p}) + pF (ℤ_{2 q}) \\ - F (ℤ_{p}) - pF (ℤ_{2}) - p \\ - ((\begin{matrix} p \\ 2 \end{matrix}) + p + 1) F (ℤ_{q}) \\ + \sum_{i = 3}^{p} (- 1)^{i - 1} (\begin{matrix} p + 1 \\ i - 1 \end{matrix}) \\ + \sum_{i = 3}^{p} (- 1)^{i - 1} (\begin{matrix} p + 1 \\ i \end{matrix}) F (ℤ_{q}) . \end{matrix}$

Thus, by Corollary 2.3 and Theorem 2.7, F (G) =10p + 16.

Then, we get the following theorem:

Theorem 3.1. The number of all distinct fuzzy subgroups of $D_{2 p} \times ℤ_{q}$ , where p, q are distinct prime numbers and p ≠ 2 is given the following equality: $F (D_{2 p} \times ℤ_{q}) = {\begin{matrix} 16 p + 20, & q = 2 \\ 10 p + 16, & q \neq 2 . \end{matrix}$

Furthermore, $F (D_{2 p} \times ℤ_{p}) = 14 p + 12$

Theorem 3.2. The number of all distinct fuzzy normal subgroups of $D_{2 p} \times ℤ_{q}$ , where p, q are prime numbers and p ≠ 2 is given the following equality: $F_{N} (D_{2 p} \times ℤ_{q}) = {\begin{matrix} 20, & q = 2 \\ 16, & {otherwise \end{matrix} .$

Proof. We prove this statement in three cases.

Case 1: q = 2. In this case, normal subgroups of G are as $M_{1} = 〈 (x, 1) 〉 ≅ ℤ_{pq}$ , M₂ = 〈 (x, 0) ; (y, 0) 〉 ≅ D_2p, $M_{3} = 〈 (x, 0); (y, 1) 〉 ≅ D_{2 p}, 〈 (x, 0) 〉 ≅ ℤ_{p}$ and $〈 (e, 1) 〉 ≅ ℤ_{2}$ . Therefore, every chain of normal subgroups of G ended in G is one of the following types: $\begin{array}{l} {(e, 0)} \subset G; G; \\ {(e, 0)} \subset M_{i} \subset G, 1 \leq i \leq 3; M_{i} \subset G, 1 \leq i \leq 3; \\ {(e, 0)} \subset 〈 (x, 0) 〉 \subset M_{i} \subset G, 1 \leq i \leq 3; \\ 〈 (x, 0) 〉 \subset M_{i} \subset G, 1 \leq i \leq 3; \\ {(e, 0)} \subset 〈 (e, 1) 〉 \subset M_{1} \subset G; 〈 (e, 1) 〉 \subset M_{1} \subset G; \\ {(e, 0)} \subset 〈 (x, 0) 〉 \subset G; 〈 (x, 0) 〉 \subset G; \\ {(e, 0)} \subset 〈 (e, 1) 〉 \subset G; 〈 (e, 1) 〉 \subset G . \end{array}$

Thus, with counting the number of the above chains, we find that the number of all chains of normal subgroups of G that terminate in G is 20. This means that $F_{N} (D_{2 p} \times ℤ_{2}) = 20$ .

Case 2: p = q. In this case, normal subgroups of G are as $M_{1} = 〈 (x, 0); (x, 1) 〉 ≅ ℤ_{p} \times ℤ_{p}$ , M₂ = 〈 (x, 0) ; (y, 0) 〉 ≅ D_2p, $〈 (x, 0) 〉 ≅ ℤ_{p}$ and $〈 (e, 1) 〉 ≅ ℤ_{2}$ . Therefore, every chain of normal subgroups of G ended in G is one of the following types: $\begin{array}{l} {(e, 0)} \subset G; G; \\ {(e, 0)} \subset M_{i} \subset G, 1 \leq i \leq 2; M_{i} \subset G, 1 \leq i \leq 2; \\ {(e, 0)} \subset 〈 (x, 0) 〉 \subset M_{i} \subset G, 1 \leq i \leq 2; \\ 〈 (x, 0) 〉 \subset M_{i} \subset G, 1 \leq i \leq 2; \\ {(e, 0)} \subset 〈 (e, 1) 〉 \subset M_{1} \subset G; 〈 (e, 1) 〉 \subset M_{1} \subset G; \\ {(e, 0)} \subset 〈 (x, 0) 〉 \subset G; 〈 (x, 0) 〉 \subset G; \\ {(e, 0)} \subset 〈 (e, 1) 〉 \subset G; 〈 (e, 1) 〉 \subset G . \end{array}$

Thus, with counting the number of the above chains, we find that the number of all chains of normal subgroups of G that terminate in G is 16. This means that $F_{N} (D_{2 p} \times ℤ_{p}) = 16$ .

Case 3: q is an odd prime number and p ≠ q. In this case, normal subgroups of G are as $M_{1} = 〈 (x, 1) 〉 ≅ ℤ_{pq}$ , M₂ = 〈 (x, 0) ; (y, 0) 〉 ≅ D_2p, $〈 (x, 0) 〉 ≅ ℤ_{p}$ and $〈 (e, 1) 〉 ≅ ℤ_{2}$ . Therefore, every chain of normal subgroups of G ended in G is one of the following types: $\begin{array}{l} {(e, 0)} \subset G; G; \\ {(e, 0)} \subset M_{i} \subset G, 1 \leq i \leq 2; M_{i} \subset G, 1 \leq i \leq 2; \\ {(e, 0)} \subset 〈 (x, 0) 〉 \subset M_{i} \subset G, 1 \leq i \leq 2; \\ 〈 (x, 0) 〉 \subset M_{i} \subset G, 1 \leq i \leq 2; \\ {(e, 0)} \subset 〈 (e, 1) 〉 \subset M_{1} \subset G; 〈 (e, 1) 〉 \subset M_{1} \subset G; \\ {(e, 0)} \subset 〈 (x, 0) 〉 \subset G; 〈 (x, 0) 〉 \subset G; \\ {(e, 0)} \subset 〈 (e, 1) 〉 \subset G; 〈 (e, 1) 〉 \subset G . \end{array}$

Thus, with counting the number of the above chains, we find that the number of all chains of normal subgroups of G that terminate in G is 16. This means that $F_{N} (D_{2 p} \times ℤ_{q}) = 16$ . □

4 Counting fuzzy (normal) subgroups of finite groups of order n ≤ 20

In this section, we determine the number of distinct fuzzy (normal) subgroups of all finite groups of order n ≤ 20.

Let $G = ℤ_{4} \times ℤ_{2} = 〈 (1, 0); (0, 1) 〉$ . The maximal subgroups of G are as follows: $\begin{matrix} M_{1} = 〈 (1, 0) 〉 ≅ ℤ_{4}; \\ M_{2} = 〈 (1, 1) 〉 ≅ ℤ_{4}; \\ M_{3} = 〈 (0, 1); (2, 0) 〉 ≅ ℤ_{2} \times ℤ_{2} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(0, 0)} \subseteq 〈 (2, 0) 〉 \subseteq M_{i} \subseteq G, 1 \leq i \leq 3; \\ {(0, 0)} \subseteq 〈 (0, 1) 〉 \subseteq M_{3} \subseteq G; \\ {(0, 0)} \subseteq 〈 (2, 1) 〉 \subseteq M_{3} \subseteq G . \end{matrix}$

We have $M_{i} \cap M_{j} = 〈 (2, 0) 〉 ≅ ℤ_{2}, 1 \leq i, j \leq 3, i \neq j,$ and $M_{1} \cap M_{2} \cap M_{3} = 〈 (2, 0) 〉 ≅ ℤ_{2}$ . Then,by Corollary 2.3 and Theorems 2.5, 2.6; $\begin{matrix} F (ℤ_{4} \times ℤ_{2}) & = & 2 (2 F (ℤ_{4}) + F (ℤ_{2} \times ℤ_{2}) - 3 F (ℤ_{2}) \\ + F (ℤ_{2})) = 24 \end{matrix}$

4.1 The number of fuzzy subgroups of abelian groups of order 16

There are five abelian groups of order 16. They are $ℤ_{16}$ , $ℤ_{4} \times ℤ_{4}$ , $ℤ_{8} \times ℤ_{2}$ , $ℤ_{4} \times ℤ_{2} \times ℤ_{2}$ and $ℤ_{2}^{4}$ . The number of fuzzy subgroups of the group $ℤ_{16}$ is obtained by Corollary 2.3. At present, we determine this number for others abelian group of order 16.

4.1.1 The group $ℤ_{4} \times ℤ_{4}$

Let $G = ℤ_{4} \times ℤ_{4} = 〈 (1, 0); (0, 1) 〉$ . The maximal subgroups of G are as follows: $\begin{matrix} M_{1} = 〈 (0, 1); (2, 2) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{2} = 〈 (1, 0); (2, 2) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{3} = 〈 (1, 1); (2, 0) 〉 ≅ ℤ_{4} \times ℤ_{2} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(0, 0)} \subseteq 〈 (2, 0) 〉 \subseteq 〈 (2, 0); (0, 2) 〉 \subseteq M_{i} \subseteq G; \\ {(0, 0)} \subseteq 〈 (0, 2) 〉 \subseteq 〈 (2, 0); (0, 2) 〉 \subseteq M_{i} \subseteq G; \\ {(0, 0)} \subseteq 〈 (2, 2) 〉 \subseteq 〈 (2, 0); (0, 2) 〉 \subseteq M_{i} \subseteq G; \\ {(0, 0)} \subseteq 〈 (0, 2) 〉 \subseteq 〈 (0, 1) 〉 \subseteq M_{1} \subseteq G; \\ {(0, 0)} \subseteq 〈 (0, 2) 〉 \subseteq 〈 (2, 1) 〉 \subseteq M_{1} \subseteq G; \\ {(0, 0)} \subseteq 〈 (2, 0) 〉 \subseteq 〈 (1, 0) 〉 \subseteq M_{2} \subseteq G; \\ {(0, 0)} \subseteq 〈 (2, 0) 〉 \subseteq 〈 (1, 2) 〉 \subseteq M_{2} \subseteq G; \\ {(0, 0)} \subseteq 〈 (2, 2) 〉 \subseteq 〈 (1, 1) 〉 \subseteq M_{3} \subseteq G; \\ {(0, 0)} \subseteq 〈 (2, 2) 〉 \subseteq 〈 (1, 3) 〉 \subseteq M_{3} \subseteq G; \end{matrix}$ where 1 ≤ i ≤ 3. We have $\begin{matrix} M_{1} \cap M_{2} & = & M_{1} \cap M_{3} = M_{2} \cap M_{3} \\ = & M_{1} \cap M_{2} \cap M_{3} \\ = & 〈 (2, 0); (0, 2) 〉 ≅ ℤ_{2} \times ℤ_{2} . \end{matrix}$

Then, by Theorems 2.5 and 2.6, $\begin{matrix} F (ℤ_{4} \times ℤ_{4}) & = & 2 (3 F (ℤ_{4} \times ℤ_{2}) - 2 F (ℤ_{2} \times ℤ_{2})) \\ = & 112 . \end{matrix}$

4.1.2 The group $ℤ_{8} \times ℤ_{2}$

Let $G = ℤ_{8} \times ℤ_{2} = 〈 (1, 0); (0, 1) 〉$ . The maximal subgroups of G are as follows: $\begin{matrix} M_{1} = 〈 (1, 0) 〉 ≅ ℤ_{8}; \\ M_{2} = 〈 (1, 1) 〉 ≅ ℤ_{8}; \\ M_{3} = 〈 (0, 1); (2, 0) 〉 ≅ ℤ_{4} \times ℤ_{2} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(0, 0)} \subseteq 〈 (4, 0) 〉 \subseteq 〈 (2, 0) 〉 \subseteq M_{i} \subseteq G, 1 \leq i \leq 3; \\ {(0, 0)} \subseteq 〈 (4, 0) 〉 \subseteq 〈 (2, 1) 〉 \subseteq M_{3} \subseteq G; \\ {(0, 0)} \subseteq 〈 (0, 1) 〉 \subseteq 〈 (0, 1); (4, 0) 〉 \subseteq M_{3} \subseteq G; \\ {(0, 0)} \subseteq 〈 (4, 0) 〉 \subseteq 〈 (0, 1); (4, 0) 〉 \subseteq M_{3} \subseteq G; \\ {(0, 0)} \subseteq 〈 (4, 1) 〉 \subseteq 〈 (0, 1); (4, 0) 〉 \subseteq M_{3} \subseteq G . \end{matrix}$

We have $\begin{matrix} M_{1} \cap M_{2} & = & M_{1} \cap M_{3} = M_{2} \cap M_{3} \\ = & M_{1} \cap M_{2} \cap M_{3} \\ = & 〈 (2, 0) 〉 ≅ ℤ_{4} . \end{matrix}$

Then, by Corollary 2.3 and Theorem 2.6, $\begin{matrix} F (ℤ_{8} \times ℤ_{2}) \\ = 2 (2 F (ℤ_{8}) + F (ℤ_{4} \times ℤ_{2}) - 2 F (ℤ_{4})) = 64 . \end{matrix}$

4.1.3 The group $ℤ_{4} \times ℤ_{2} \times ℤ_{2}$

Let $G = ℤ_{4} \times ℤ_{2} \times ℤ_{2} = < (1, 0, 0); (0, 1, 0); (0, 0, 1) >$ . The maximal subgroups of G are as follows: $\begin{matrix} M_{1} = 〈 (1, 0, 0); (0, 0, 1) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{2} = 〈 (1, 0, 0); (0, 1, 0) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{3} = 〈 (1, 0, 0); (0, 1, 1) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{4} = 〈 (1, 0, 1); (0, 1, 1) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{5} = 〈 (1, 0, 1); (0, 1, 0) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{6} = 〈 (1, 1, 0); (0, 0, 1) 〉 ≅ ℤ_{4} \times ℤ_{2}; \\ M_{7} = 〈 (0, 0, 1); (0, 1, 0); (0, 1, 1) 〉 \\ ≅ ℤ_{2} \times ℤ_{2} \times ℤ_{2} . \end{matrix}$

If we set (0, 0, 0) : = e, then the maximal chains of G are one of the following types: $\begin{matrix} {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (1, 0, 0) 〉 \\ \subseteq & M_{i} \subseteq G, i = 1, 2, 3; \\ {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (1, 0, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 1, 4, 5; \\ {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (1, 1, 0) 〉 \\ \subseteq & M_{i} \subseteq G, i = 2, 4, 6; \\ {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (1, 1, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 3, 5, 6; \\ {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (2, 0, 0), (2, 0, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 1, 6, 7; \\ {e} & \subseteq & 〈 (2, 0, 1) 〉 \subseteq 〈 (2, 0, 0), (2, 0, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 1, 6, 7; \\ {e} & \subseteq & 〈 (0, 0, 1) 〉 \subseteq 〈 (2, 0, 0), (2, 0, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 1, 6, 7; \\ {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (2, 0, 0), (2, 1, 0) 〉 \\ \subseteq & M_{i} \subseteq G, i = 2, 5, 7; \\ {e} & \subseteq & 〈 (2, 1, 0) 〉 \subseteq 〈 (2, 0, 0), (2, 1, 0) 〉 \\ \subseteq & M_{i} \subseteq G, i = 2, 5, 7; \\ {e} & \subseteq & 〈 (0, 1, 0) 〉 \subseteq 〈 (2, 0, 0), (2, 1, 0) 〉 \\ \subseteq & M_{i} \subseteq G, i = 2, 5, 7; \\ {e} & \subseteq & 〈 (2, 0, 0) 〉 \subseteq 〈 (2, 0, 0), (2, 1, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 3, 4, 7; \\ {e} & \subseteq & 〈 (2, 1, 1) 〉 \subseteq 〈 (2, 0, 0), (2, 1, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 3, 4, 7; \\ {e} & \subseteq & 〈 (0, 1, 1) 〉 \subseteq 〈 (2, 0, 0), (2, 1, 1) 〉 \\ \subseteq & M_{i} \subseteq G, i = 3, 4, 7 . \end{matrix}$

We find that $\begin{matrix} M_{1} \cap M_{2} & = & M_{1} \cap M_{3} = M_{2} \cap M_{3} \\ = & M_{1} \cap M_{2} \cap M_{3} = 〈 (1, 0, 0) 〉 ≅ ℤ_{4}; \\ M_{1} \cap M_{4} & = & M_{1} \cap M_{5} = M_{4} \cap M_{5} \\ = & M_{1} \cap M_{4} \cap M_{5} = 〈 (1, 0, 1) 〉 ≅ ℤ_{4}; \\ M_{2} \cap M_{4} & = & M_{2} \cap M_{6} = M_{4} \cap M_{6} \\ = & M_{2} \cap M_{4} \cap M_{6} = 〈 (1, 1, 0) 〉 ≅ ℤ_{4}; \\ M_{3} \cap M_{5} & = & M_{3} \cap M_{6} = M_{5} \cap M_{6} \\ = & M_{3} \cap M_{5} \cap M_{6} = 〈 (1, 1, 1) 〉 ≅ ℤ_{4}; \\ M_{1} \cap M_{6} & = & M_{1} \cap M_{7} = M_{6} \cap M_{7} \\ = & M_{1} \cap M_{6} \cap M_{7} = 〈 (2, 0, 0); (2, 0, 1) 〉 \\ ≅ & ℤ_{2} \times ℤ_{2}; \\ M_{2} \cap M_{5} & = & M_{2} \cap M_{7} = M_{5} \cap M_{7} \\ = & M_{2} \cap M_{5} \cap M_{7} = 〈 (2, 0, 0); (2, 1, 0) 〉 \\ ≅ & ℤ_{2} \times ℤ_{2}; \\ M_{3} \cap M_{4} & = & M_{3} \cap M_{7} = M_{4} \cap M_{7} \\ = & M_{3} \cap M_{4} \cap M_{7} = 〈 (2, 0, 0); (2, 1, 1) 〉 \\ ≅ & ℤ_{2} \times ℤ_{2} . \end{matrix}$

Other intersections of subgroups is the group $〈 (2, 0, 0) 〉 ≅ ℤ_{2}$ . Then, by Corollary 2.3 and Theorems 2.5, 2.6, $\begin{matrix} F (ℤ_{4} \times ℤ_{2} \times ℤ_{2}) \\ = 2 (6 F (ℤ_{4} \times ℤ_{2}) + F (ℤ_{2} \times ℤ_{2} \times ℤ_{2}) \\ - 8 F (ℤ_{4}) - 6 F (ℤ_{2} \times ℤ_{2}) \\ + 8 F (ℤ_{2})) = 304 . \end{matrix}$

4.1.4 The group $ℤ_{2}^{4}$

Let $G = ℤ_{2}^{4}$ and H₁, H₂, …, H₁₅ be minimal subgroups of G. Then, for all r ∈ {1, 2, …, 15} and 1 ≤ s ≤ min {r, 4}, we have $\begin{matrix} x_{r, s} & = & | {(i_{1}, i_{2}, \dots, i_{r}) | 1 \leq i_{1} < i_{2} \\ < \dots < i_{r} \leq 15; | H_{i_{1}} + H_{i_{2}} \\ + \dots + H_{i_{r}} | = 2^{s}} | . \end{matrix}$

In this case, the coefficients x_r,s in the recurrence relation of Theorem 2.5 have the next values: $\begin{matrix} x_{1, 1} = 15; x_{i, 1} = 0, 2 \leq i \leq 15; \\ x_{2, 2} = (\begin{matrix} 15 \\ 2 \end{matrix}); x_{3, 2} = (\begin{matrix} 7 \\ 3 \end{matrix}) = 35; \\ x_{i, 2} = 0, 4 \leq i \leq 15; x_{3, 3} = (\begin{matrix} 15 \\ 3 \end{matrix}) - 35; \\ x_{i, 3} = 15 (\begin{matrix} 7 \\ i \end{matrix}), 4 \leq i \leq 7; x_{i, 3} = 0, 8 \leq i \leq 15; \\ x_{i, 4} = (\begin{matrix} 15 \\ i \end{matrix}) - x_{i, 3}, 4 \leq i \leq 15 . \end{matrix}$

So, by Corollary 2.3 and Theorem 2.5, $\begin{matrix} F (ℤ_{2}^{4}) \\ = 2 \sum_{r = 1}^{15} (- 1)^{r - 1} \sum_{s = 1}^{min {r, 4}} x_{r, s} F (ℤ_{2}^{4 - s}) \\ = 2 {15 F (ℤ_{2}^{3}) + ((\begin{matrix} 7 \\ 3 \end{matrix}) - (\begin{matrix} 15 \\ 2 \end{matrix})) F (ℤ_{2}^{2}) \\ + 15 (- (\begin{matrix} 7 \\ 4 \end{matrix}) + (\begin{matrix} 7 \\ 5 \end{matrix}) - (\begin{matrix} 7 \\ 6 \end{matrix}) + (\begin{matrix} 7 \\ 7 \end{matrix})) F (ℤ_{2}) \\ - (\begin{matrix} 15 \\ 4 \end{matrix}) + (\begin{matrix} 15 \\ 5 \end{matrix}) - (\begin{matrix} 15 \\ 6 \end{matrix}) + \dots + (\begin{matrix} 15 \\ 5 \end{matrix}) \\ - 15 (- (\begin{matrix} 7 \\ 4 \end{matrix}) + (\begin{matrix} 7 \\ 5 \end{matrix}) - (\begin{matrix} 7 \\ 6 \end{matrix}) + (\begin{matrix} 7 \\ 7 \end{matrix})) \\ + ((\begin{matrix} 15 \\ 3 \end{matrix}) - 35) F (ℤ_{2})} = 1392 . \end{matrix}$

4.2 The number of fuzzy (normal) subgroups of some groups of order 18

4.2.1 The group $ℤ_{3} \times ℤ_{3} \times ℤ_{2}$

Let $G = ℤ_{3} \times ℤ_{3} \times ℤ_{2} = < (1, 0, 0); (0, 1, 0); (0, 0, 1) >$ . The maximal subgroups of G are as follows: $\begin{matrix} M_{1} = 〈 (0, 1, 1) 〉 ≅ ℤ_{6}; M_{2} = 〈 (1, 0, 1) 〉 ≅ ℤ_{6}; \\ M_{3} = 〈 (1, 1, 1) 〉 ≅ ℤ_{6}; M_{4} = 〈 (1, 2, 1) 〉 ≅ ℤ_{6}; \\ M_{5} = 〈 (0, 1, 0); (1, 0, 0) 〉 ≅ ℤ_{3} \times ℤ_{3} . \end{matrix}$

Maximal chains of G are one of the following types: $\begin{matrix} {(0, 0, 0)} \subseteq 〈 (0, 0, 1) 〉 \subseteq M_{i} \subseteq G, 1 \leq i \leq 4; \\ {(0, 0, 0)} \subseteq 〈 (0, 1, 0) 〉 \subseteq M_{i} \subseteq G, i = 1, 5; \\ {(0, 0, 0)} \subseteq 〈 (1, 0, 0) 〉 \subseteq M_{i} \subseteq G, i = 2, 5; \\ {(0, 0, 0)} \subseteq 〈 (1, 1, 0) 〉 \subseteq M_{i} \subseteq G, i = 3, 5; \\ {(0, 0, 0)} \subseteq 〈 (2, 1, 0) 〉 \subseteq M_{i} \subseteq G, i = 4, 5 . \end{matrix}$

We find that $\begin{matrix} M_{1} \cap M_{i} = 〈 (0, 0, 1) 〉 ≅ ℤ_{2}, i = 2, 3, 4; \\ M_{2} \cap M_{i} = 〈 (0, 0, 1) 〉 ≅ ℤ_{2}, i = 3, 4; \\ M_{1} \cap M_{5} = 〈 (0, 1, 0) 〉 ≅ ℤ_{3}; \\ M_{2} \cap M_{5} = 〈 (1, 0, 0) 〉 ≅ ℤ_{3}; \\ M_{3} \cap M_{4} = 〈 (0, 0, 1) 〉 ≅ ℤ_{2}; \\ M_{3} \cap M_{5} = 〈 (1, 1, 0) 〉 ≅ ℤ_{3}; \\ M_{4} \cap M_{5} = 〈 (2, 1, 0) 〉 ≅ ℤ_{3}; \end{matrix}$ and $\begin{matrix} M_{1} \cap M_{2} \cap M_{3} \\ = M_{1} \cap M_{2} \cap M_{4} = M_{1} \cap M_{3} \cap M_{4} \\ = M_{2} \cap M_{3} \cap M_{4} \\ = M_{1} \cap M_{2} \cap M_{3} \cap M_{4} \\ = 〈 (0, 0, 1) 〉 ≅ ℤ_{2} . \end{matrix}$

Other intersections of subgroups are the trivial group 〈(0, 0, 0) 〉. Then, by Corollary 2.3 and Theorems 2.5, 2.6, $\begin{matrix} F (ℤ_{3} \times ℤ_{3} \times ℤ_{2}) \\ = 2 (4 F (ℤ_{6}) + F (ℤ_{3} \times ℤ_{3}) - 3 F (ℤ_{2}) \\ - 4 F (ℤ_{3}) + 3) = 46 . \end{matrix}$

4.2.2 The group $(ℤ_{3} \times ℤ_{3}) ⋉ ℤ_{2}$

Let $G = (ℤ_{3} \times ℤ_{3}) ⋉ ℤ_{2}$ , i.e., G = < x, y, z |x² = y³ = z³ = e ; yz = zy, x^-1yx = y^-1, x^-1zx = z^-1>. Then, G has 8 elements of order 3 and 9 elements of order 2. The maximal subgroups of G are as follows: $\begin{matrix} M_{1} = 〈 y, z 〉; M_{2} = 〈 x, y 〉; M_{3} = 〈 {xz}^{2}, y 〉; \\ M_{4} = 〈 xz, y 〉; M_{5} = 〈 x, z 〉; M_{6} = 〈 {xy}^{2}, z 〉; \\ M_{7} = 〈 xy, z 〉; M_{8} = 〈 x, {yz}^{2} 〉; M_{9} = 〈 {xz}^{2}, {yz}^{2} 〉; \\ M_{10} = 〈 xz, {yz}^{2} 〉; M_{11} = 〈 x, yz 〉; \\ M_{12} = 〈 {xz}^{2}, yz 〉; M_{13} = 〈 xz, yz 〉 . \end{matrix}$

Then, $M_{1} ≅ ℤ_{3} \times ℤ_{3}$ and M_i ≅ D₆, for all 2 ≤ i ≤ 13. Maximal chains of G are one of the following types: $\begin{matrix} {e} \subseteq 〈 y 〉 \subseteq M_{i} \subseteq G, i = 1, 2, 3, 4; \\ {e} \subseteq 〈 z 〉 \subseteq M_{i} \subseteq G, i = 1, 5, 6, 7; \\ {e} \subseteq 〈 {yz}^{2} 〉 \subseteq M_{i} \subseteq G, i = 1, 8, 9, 10; \\ {e} \subseteq 〈 yz 〉 \subseteq M_{i} \subseteq G, i = 1, 11, 12, 13; \\ {e} \subseteq 〈 x 〉 \subseteq M_{i} \subseteq G, i = 2, 5, 8, 11; \\ {e} \subseteq 〈 xy 〉 \subseteq M_{i} \subseteq G, i = 2, 7, 10, 12; \\ {e} \subseteq 〈 {xy}^{2} 〉 \subseteq M_{i} \subseteq G, i = 2, 6, 9, 13; \\ {e} \subseteq 〈 xz 〉 \subseteq M_{i} \subseteq G, i = 4, 5, 10, 13; \\ {e} \subseteq 〈 {xz}^{2} 〉 \subseteq M_{i} \subseteq G, i = 3, 5, 9, 12; \\ {e} \subseteq 〈 xyz 〉 \subseteq M_{i} \subseteq G, i = 4, 7, 9, 11; \\ {e} \subseteq 〈 {xy}^{2} z 〉 \subseteq M_{i} \subseteq G, i = 4, 6, 8, 12; \\ {e} \subseteq 〈 {xyz}^{2} 〉 \subseteq M_{i} \subseteq G, i = 3, 7, 8, 13; \\ {e} \subseteq 〈 {xy}^{2} z^{2} 〉 \subseteq M_{i} \subseteq G, i = 3, 6, 10, 11 . \end{matrix}$

We have $\begin{matrix} M_{i} \cap M_{j} = 〈 y 〉 ≅ ℤ_{3}, i \neq j, i, j = 1, 2, 3, 4; \\ M_{i} \cap M_{j} = 〈 z 〉 ≅ ℤ_{3}, i \neq j, i, j = 1, 5, 6, 7; \\ M_{i} \cap M_{j} = 〈 {yz}^{2} 〉 ≅ ℤ_{3}, i \neq j, i, j = 1, 8, 9, 10; \\ M_{i} \cap M_{j} = 〈 yz 〉 ≅ ℤ_{3}, i \neq j, i, j = 1, 11, 12, 13; \\ M_{i} \cap M_{j} = 〈 x 〉 ≅ ℤ_{2}, i \neq j, i, j = 2, 5, 8, 11; \\ M_{i} \cap M_{j} = 〈 xy 〉 ≅ ℤ_{2}, i \neq j, i, j = 2, 7, 10, 12; \\ M_{i} \cap M_{j} = 〈 {xy}^{2} 〉 ≅ ℤ_{2}, i \neq j, i, j = 2, 6, 9, 13; \\ M_{i} \cap M_{j} = 〈 {xz}^{2} 〉 ≅ ℤ_{2}, i \neq j, i, j = 3, 5, 9, 12; \\ M_{i} \cap M_{j} = 〈 {xy}^{2} z^{2} 〉 ≅ ℤ_{2}, i \neq j, \\ i, j = 3, 6, 10, 11; \\ M_{i} \cap M_{j} = 〈 {xyz}^{2} 〉 ≅ ℤ_{2}, i \neq j, i, j = 3, 7, 8, 13; \\ M_{i} \cap M_{j} = 〈 xz 〉 ≅ ℤ_{2}, i \neq j, i, j = 4, 5, 10, 13; \\ M_{i} \cap M_{j} = 〈 {xy}^{2} z 〉 ≅ ℤ_{2}, i \neq j, i, j = 4, 6, 8, 12; \\ M_{i} \cap M_{j} = 〈 xyz 〉 ≅ ℤ_{2}, i \neq j, i, j = 4, 7, 9, 11 . \end{matrix}$

Other intersections of subgroups are the trivial group 〈(0, 0, 0) 〉. Then, by Corollary 2.3 and Theorems 2.5, 2.6, 2.7, $\begin{matrix} (1 / 2) F (G) \\ = F (ℤ_{3} \times ℤ_{3}) + 12 F (D_{6}) - 12 F (ℤ_{3}) \\ - 27 F (ℤ_{2}) + (\begin{matrix} 13 \\ 3 \end{matrix}) - 52 - (\begin{matrix} 13 \\ 4 \end{matrix}) + 13 \\ + (\begin{matrix} 13 \\ 5 \end{matrix}) - (\begin{matrix} 13 \\ 6 \end{matrix}) + \dots + (\begin{matrix} 13 \\ 13 \end{matrix}) = 79 . \end{matrix}$

So, F (G) =158. The normal subgroups of G are as $M_{1} = 〈 y, z 〉 ≅ ℤ_{3} \times ℤ_{3}$ , $H_{1} = 〈 y 〉 ≅ ℤ_{3}$ , $H_{2} = 〈 z 〉 ≅ ℤ_{3}$ , $H_{3} = 〈 yz 〉 ≅ ℤ_{3}$ and $H_{4} = 〈 {yz}^{2} 〉 ≅ ℤ_{3}$ . Therefore, every chain of normal subgroups of G ended in G is one of the following types: $\begin{matrix} {e} \subset G; G; \\ {e} \subset M_{1} \subset G; M_{1} \subseteq G; \\ {e} \subset H_{i} \subseteq M_{1} \subset G, 1 \leq i \leq 4; \\ H_{i} \subset M_{1} \subset G, 1 \leq i \leq 4; \\ {e} \subset H_{i} \subset G, 1 \leq i \leq 4; H_{i} \subset G, 1 \leq i \leq 4 . \end{matrix}$

Thus, with counting the number of the above chains, we find that the number of all chains of normal subgroups of G that terminate in G is 20. This means that F_N (G) =20.

4.3 Determining the number of fuzzy (normal) subgroups of all finite groups of order n ≤ 20

Now, we can classify the fuzzy (normal) subgroups of all finite groups of order n ≤ 20. The number of distinct fuzzy subgroups and fuzzy normal subgroups of these groups is given in Tables 1 and 2.

Table 1

The number of distinct fuzzy subgroups of all finite groups of order n ≤ 20

Order	Group	F (G)	Reference
1	{0}	1	Theorem 2.6
2	$ℤ_{2}$	2	Corollary 2.3
3	$ℤ_{3}$	2	Corollary 2.3
4	$ℤ_{4}$	4	Corollary 2.3
	$ℤ_{2} \times ℤ_{2}$	8	Theorem 2.5
5	$ℤ_{5}$	2	Corollary 2.3
6	$ℤ_{6}$	6	Corollary 2.3
	D₆ = 〈x, y \| x³ = e, y² = e, y^-1xy = x^-1〉	10	Theorem 2.7
7	$ℤ_{7}$	2	Corollary 2.3
8	$ℤ_{8}$	8	Corollary 2.3
	$ℤ_{2} \times ℤ_{2} \times ℤ_{2}$	72	Theorem 2.5
	$ℤ_{4} \times ℤ_{2}$	24	Section 4
	D₈ = 〈x, y \| x⁴ = e, y² = e, y^-1xy = x^-1〉	32	Theorem 2.7
	T₈ = Q₈ = 〈x, y \| x⁴ = e, y² = x², y^-1xy = x^-1〉	16	Theorem 2.8
9	$ℤ_{9}$	4	Corollary 2.3
	$ℤ_{3} \times ℤ_{3}$	10	Theorem 2.5
10	$ℤ_{10}$	6	Corollary 2.3
	D₁₀ = 〈x, y \| x⁵ = e, y² = e, y^-1xy = x^-1〉	14	Theorem 2.7
11	$ℤ_{11}$	2	Corollary 2.3
12	$ℤ_{12}$	16	Corollary 2.3
	$ℤ_{2} \times ℤ_{2} \times ℤ_{3}$	40	Theorem 2.4
	D₁₂ = 〈x, y \| x⁶ = e, y² = e, y^-1xy = x^-1〉	68	Theorem 2.7
	T₁₂ = 〈x, y \| x⁶ = e, y² = x³, y^-1xy = x^-1〉	24	Theorem 2.8
	$D_{6} \times ℤ_{2}$	68	Theorem 3.1
13	$ℤ_{13}$	2	Corollary 2.3
14	$ℤ_{14}$	6	Corollary 2.3
	D₁₄ = 〈x, y \| x⁷ = e, y² = e, y^-1xy = x^-1〉	18	Theorem 2.7
15	$ℤ_{15}$	6	Corollary 2.3
16	$ℤ_{16}$	16	Corollary 2.3
	$ℤ_{4} \times ℤ_{4}$	112	Subsection 4.1
	$ℤ_{8} \times ℤ_{2}$	64	Subsection 4.1
	$ℤ_{4} \times ℤ_{2} \times ℤ_{2}$	304	Subsection 4.1
	$ℤ_{2} \times ℤ_{2} \times ℤ_{2} \times ℤ_{2}$	1392	Subsection 4.1
	D₁₆ = 〈x, y \| x⁸ = e, y² = e, y^-1xy = x^-1〉	2⁷	Theorem 2.7
	T₁₆ = 〈x, y \| x⁸ = e, y² = x⁴, y^-1xy = x^-1〉	4³	Theorem 2.8
	〈x, y \| x⁸ = y² = e, y^-1xy = x³〉	96	[4, Section 4]
	〈x, y \| x⁸ = y² = e, y^-1xy = x⁵〉	64	[4, Section 4]
	〈x, y \| x⁴ = y⁴ = e, y^-1xy = xy²〉	112	[4, Section 4]
	〈x, y, z \| x⁴ = y² = z² = e, y^-1xy = xz, xz = zx, yz = zy〉	208	[4, Section 4]
	$D_{8} \times ℤ_{2}$	432	[4, Section 4]
	$Q_{8} \times ℤ_{2}$	176	[4, Section 4]
	〈x, y, z \| x² = y² = z⁴ = e, y^-1xy = xz², xz = zx, yz = zy〉	240	[4, Section 4]
17	$ℤ_{17}$	2	Corollary 2.3
18	$ℤ_{18}$	16	Corollary 2.3
	$ℤ_{3} \times ℤ_{3} \times ℤ_{2}$	46	Subsection 4.2
	D₁₈ = 〈x, y \| x⁹ = e, y² = e, y^-1xy = x^-1〉	20	Theorem 2.7
	$D_{6} \times ℤ_{3}$	54	Theorem 3.1
	〈x, y, z \| x² = y³ = z³ = e ; yz = zy, x^-1yx = y^-1, x^-1zx = z^-1〉	158	Subsection 4.2
19	$ℤ_{19}$	2	Corollary 2.3
20	$ℤ_{20}$	16	Corollary 2.3
	$ℤ_{2} \times ℤ_{2} \times ℤ_{5}$	40	Theorem 2.4
	D₂₀ = 〈x, y \| x¹⁰ = e, y² = e, y^-1xy = x^-1〉	100	Theorem 2.7
	T₂₀ = 〈x, y \| x¹⁰ = e, y² = x⁵, y^-1xy = x^-1〉	32	Theorem 2.8
	$D_{10} \times ℤ_{2}$	100	Theorem 3.1

Table 2

The number of distinct fuzzy normal subgroups of all finite nonabelian groups of order n ≤ 20

Order	Group	F_N (G)	Reference
6	D₆ = 〈x, y \| x³ = e, y² = e, y^-1xy = x^-1〉	4	Theorem 2.7
8	D₈ = 〈x, y \| x⁴ = e, y² = e, y^-1xy = x^-1〉	16	Theorem 2.7
	T₈ = Q₈	16	Theorem 2.8
10	D₁₀ = 〈x, y \| x⁵ = e, y² = e, y^-1xy = x^-1〉	4	Theorem 2.7
12	D₁₂ = 〈x, y \| x⁶ = e, y² = e, y^-1xy = x^-1〉	24	Theorem 2.7
	T₁₂ = 〈x, y \| x⁶ = e, y² = x³, y^-1xy = x^-1〉	12	Theorem 2.8
	$D_{6} \times ℤ_{2}$	20	Theorem 3.2
14	D₁₄ = 〈x, y \| x⁷ = e, y² = e, y^-1xy = x^-1〉	4	Theorem 2.7
16	D₁₆ = 〈x, y \| x⁸ = e, y² = e, y^-1xy = x^-1〉	32	Theorem 2.7
	T₁₆ = 〈x, y \| x⁸ = e, y² = x⁴, y^-1xy = x^-1〉	32	Theorem 2.8
	〈x, y \| x⁸ = y² = e, y^-1xy = x³〉	32	[5, Section 4]
	〈x, y \| x⁸ = y² = e, y^-1xy = x⁵〉	48	[5, Section 4]
	〈x, y \| x⁴ = y⁴ = e, y^-1xy = xy²〉	80	[5, Section 4]
	〈x, y, z \| x⁴ = y² = z² = e, y^-1xy = xz, xz = zx, yz = zy〉	80	[5, Section 4]
	$D_{8} \times ℤ_{2}$	176	[5, Section 4]
	$Q_{8} \times ℤ_{2}$	176	[5, Section 4]
	〈x, y, z \| x² = y² = z⁴ = e, y^-1xy = xz², xz = zx, yz = zy〉	144	5, [Section 4]
18	D₁₈ = 〈x, y \| x⁹ = e, y² = e, y^-1xy = x^-1〉	8	Theorem 2.7
	$D_{6} \times ℤ_{3}$	16	Theorem 3.2
	〈x, y, z \| x² = y³ = z³ = e ; yz = zy, x^-1yx = y^-1, x^-1zx = z^-1〉	20	Subsection 4.2
20	D₂₀ = 〈x, y \| x¹⁰ = e, y² = e, y^-1xy = x^-1〉	20	Theorem 2.7
	T₂₀ = 〈x, y \| x¹⁰ = e, y² = x⁵, y^-1xy = x^-1〉	12	Theorem 2.8
	$D_{10} \times ℤ_{2}$	20	Theorem 3.2

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Classifying fuzzy (normal) subgroups of the group D 2 p × ℤ q and finite groups of order n ≤ 20

Abstract

Keywords

1 Introduction

2 Preliminaries

3 The number of fuzzy (normal) subgroups of finite group D 2 p × ℤ q

4 Counting fuzzy (normal) subgroups of finite groups of order n ≤ 20

4.1 The number of fuzzy subgroups of abelian groups of order 16

4.1.1 The group ℤ 4 × ℤ 4

4.1.2 The group ℤ 8 × ℤ 2

4.1.3 The group ℤ 4 × ℤ 2 × ℤ 2

4.1.4 The group ℤ 2 4

4.2 The number of fuzzy (normal) subgroups of some groups of order 18

4.2.1 The group ℤ 3 × ℤ 3 × ℤ 2

4.2.2 The group ( ℤ 3 × ℤ 3 ) ⋉ ℤ 2

4.3 Determining the number of fuzzy (normal) subgroups of all finite groups of order n ≤ 20

References

3 The number of fuzzy (normal) subgroups of finite group $D_{2 p} \times ℤ_{q}$

4.1.1 The group $ℤ_{4} \times ℤ_{4}$

4.1.2 The group $ℤ_{8} \times ℤ_{2}$

4.1.3 The group $ℤ_{4} \times ℤ_{2} \times ℤ_{2}$

4.1.4 The group $ℤ_{2}^{4}$

4.2.1 The group $ℤ_{3} \times ℤ_{3} \times ℤ_{2}$

4.2.2 The group $(ℤ_{3} \times ℤ_{3}) ⋉ ℤ_{2}$