Abstract
FUZZY INFORMATION AND ENGINEERING https://doi.org/10.1080/16168658.2022.2119828 On the Subgroups Lattice and Fuzzy Subgroups of Finite Groups U 6n a b L. Kamali Ardekani and B. Davvaz a b Faculty of Engineering, Ardakan University, P.O. Box 184, Ardakan, Iran; Department of Mathematics, Yazd University, Yazd, Iran ABSTRACT ARTICLE HISTORY Received 3 March 2021 In this paper, we treat accounting for the number of fuzzy (normal) Revised 1 December 2021 subgroups of finite groups U . In order to do this, we use the nat- 6n Accepted 23 August 2022 ural equivalence relation on the set of fuzzy (normal) subgroups of U , which has a consistent group theoretical foundation. In fact, the 6n KEYWORDS corresponding equivalence classes of fuzzy (normal) subgroups of Equivalence relation; lattice; U are closely connected to the structure of (normal) subgroups lat- 6n fuzzy subgroup; chain of tice of U and chains of subgroups of U , which terminate in U . subgroups; level subset; 6n 6n 6n metacyclic group In this regards, the Inclusion-Exclusion principle plays an essential role and in some situations leads to recurrence relations, whose their MATHEMATICS SUBJECT solutions can be easily found. CLASSIFICATIONS (2010) 20N25; 20E15 1. Introduction In 1965, Zadeh introduced the concept of fuzzy sets as a generalisation of classical sets the- ory [1]. Exactly, a fuzzy subset μ(x) of a set X is a function from X to the unit closed interval [0, 1]. Zadeh’s ideas created new directions for researchers worldwide. Since the inception of the theory of fuzzy sets in mathematics, researchers in various disciplines of mathemat- ics were trying to extend their notions to the framework of fuzzy logic. In 1971, Rosenfeld used the fuzzy concept to develop the theory of fuzzy groups [2]. Since the notion of a fuzzy group is a generalisation of the notion of group, many basic properties in group the- ory extended to fuzzy groups. In the early 80s, the study of fuzzy normal subgroups was initiated by Wu in Ref. [3] and Mukherjee and Bhattacharya in Ref. [4]. Without any equivalence on fuzzy (normal) subgroups of a group, the number of fuzzy (normal) subgroups of a finite group is infinite even for the trivial group {e}. Therefore, some remarkable papers have treated the classification of the fuzzy subgroups for particu- lar cases of finite groups with respect to suitable equivalence relations. Also, a comparison between some equivalence relations on fuzzy (normal) subgroups lattice of a group and the behaviour of different equivalence classes is given in Refs. [5–10]. A starting point for our discussion is the natural equivalence relation ∼ introduced in Ref. [11]. This equivalence relation is closely connected to the concept of level subsets and extends the equiva- lence relation used in Murali’s papers [12–14]. By this equivalence relation, the number CONTACT L. Kamali Ardekani l.kamali@ardakan.ac.ir © 2022 The Authors. Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 L. KAMALI ARDEKANI AND B. DAVVAZ of fuzzy subgroups of certain families of finite groups is determined, for example see Refs. [10, 15–20]. In this paper, we follow to obtain the number of fuzzy subgroups of the group U with 6n respect to ∼ . In this regard, in Section 2, we present some preliminary definitions and nec- essary results on fuzzy subgroups, which we will need in the next sections. In Sections 3 and 4, we deal with counting the number of all distinct fuzzy subgroups and fuzzy normal subgroups of finite groups U , by using the equivalence relation ∼ . These numbers are 6n denoted by N (U ) and N (U ), respectively. F 6n FN 6n 2. Preliminaries In this section, we recall some necessary notions and results of fuzzy groups theory, for more details, we refer the reader to Refs. [2–4, 21]. A fuzzy subset μ of a group G is called a fuzzy subgroup if it satisfies the condi- −1 tions μ(xy) ≥ min{μ(x), μ(y)} and μ(x ) ≥ μ(x), for all x, y ∈ G. In this situation, we −1 have μ(x ) = 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, or equivalently −1 μ(xyx ) = μ(y). The set FL(G) (FN(G), respectively) consist of all fuzzy subgroups (fuzzy normal subgroups, respectively) of G forms a lattice with respect to fuzzy set inclusion. In Ref. [1], along with fuzzy sets also the concept of level subsets has been introduced as U(μ, t) ={x ∈ X | μ(x) ≥ t}, where t ∈ [0, 1]. These subsets allow us to characterise the fuzzy subgroups of a group G in this manner: A fuzzy subset μ of G is a fuzzy subgroup if and only if its non-empty level subsets are subgroups of G. In fact, level subsets per- mit one to perceive the theory of fuzzy groups within the area of classical groups theory. It has also been proved that the family of level subsets of a given fuzzy subgroup forms a chain [21]. Exactly, let G be a finite group and μ be a fuzzy subgroup of G such that Im μ ={t , t , ... , t }, where t > t > ... > t . Then, μ determines the following chain of 1 2 r 1 2 r subgroups of G, which ends in G: U(μ, t ) ⊂ U(μ, t ) ⊂ ... ⊂ U(μ, t ) = G.(1) 1 2 r Moreover, for any x ∈ G and i = 1, ... , r we have μ(x) = t ⇐⇒ i = max{j | x ∈ U(μ, t )}⇐⇒ x ∈ U(μ, t )\U(μ, t ), i j i i−1 where by convention, we set U(μ, t ) =∅. The fuzzy (normal, respectively) subgroups of G can be classified up to some natu- ral equivalence relations on FL(G) (FN(G), respectively). One of the equivalence relations related to the concept of level subsets is defined as follows: Let μ and η be two fuzzy subgroups of G. Then, we define μ ∼ η if and only if (μ(x)>μ(y) ⇐⇒ η(x)>η(y), for all x, y ∈ G) ; and two fuzzy (normal) subgroups μ and η of G will be called distinct if μ η. A necessary and sufficient condition for fuzzy subgroups μ and η of G to be equivalent with respect to ∼ is that they determine the same chain of of type (1) [11]. This result shows that there exists a bijection between the equivalence classes of fuzzy (normal) subgroups of G and the FUZZY INFORMATION AND ENGINEERING 3 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 with respect to ∼ can be translated into a combinatorial problem: Finding the number of all chains of (normal) subgroups of G that terminate in G. The largest class of groups for which the problem of counting, with respect to ∼ is completely solved is constituted by finite cyclic groups in the following manner: Theorem 2.1 ([18, Corollary 4]): If G is a finite cyclic group of order n (that is G Z )and m m m 1 2 s n = p p ... p is the decomposition of n as a product of prime factors, then the number of 1 2 all distinct fuzzy subgroups of G is given by the equality: m m m s 2 3 s α s s m m + (m − i ) α 1 β β m i β=2 α α α=1 α=2 N (Z ) = 2 ... (−1/2) , F n i m α α i =0 i =0 i =0 α=2 2 3 s and the above-iterated sums are equal to 1 for s = 1. Corollary 2.1 ([18, Remark 6]): The number of all distinct fuzzy subgroups of the finite cyclic n m group G of order p q (p, q primes) is given by the equality: n m n+m r N (G) = 2 (1/2 ) . r r r=0 In particular, the number of all distinct fuzzy subgroups of the finite cyclic p-group G (i.e. when m = 0) is 2 . The next theorem based on Inclusion-Exclusion Principle has an important role in this paper and describes the method that will be used in the calculating N (U ). F 6n Theorem 2.2 ([19, Section 2]): Let G be a group and M , M , ... , M be maximal subgroups 1 2 k of G. Then,N (G) is given by the following equality: ⎛ ⎛ ⎞ ⎞ k k k−1 ⎝ ⎝ ⎠ ⎠ N (G) = 2 N (M ) − N (M ∩ M ) + ··· + (−1) N M . F F i F i i F i 1 2 i=1 i=1 1≤i <i ≤k 1 2 If maximal subgroup structure of G is known, in some cases, Theorem 2.2 will lead to recurrence relations that permit us to determine N (G). For example, consider the group G = Z × Z =(1, 0), (0, 1) . The maximal subgroups of G are 9 3 ∼ ∼ M =(3, 0), (0, 1) = Z × Z and M =(1, i − 2) = Z , 1 3 3 i 9 where 2 ≤ i ≤ 4. Therefore, we get M ∩ M = M ∩ M ∩ M = M ∩ M ∩ M ∩ M =(3, 0) = Z , i j i j k 1 2 3 4 3 where 1 ≤ i, j, k ≤ 4and i
= j
= k. Now, Theorem 2.2 follows that N (Z × Z ) = 2 (3N (Z ) + N (Z × Z ) − 3N (Z )) . F 9 3 F 9 F 3 3 F 3 Since N (Z ) = 10, [18, Theorem 9] then by Corollary 2.1, we get the following theorem: 3 4 L. KAMALI ARDEKANI AND B. DAVVAZ Theorem 2.3: The number of all distinct fuzzy subgroups of the group Z × Z is 32. 9 3 Also by Theorem 2.2, the number of distinct fuzzy subgroups of dihedral groups D = 2n n 2 −1 −1 2n 2 n −1 −1 x, y | x = y = e, y xy = x , dicyclic groups T =x, y | x = e, y = x , y xy = x 4n and the groups D × Z are determined in Refs. [7, 16, 19], respectively. In particular, we 2p q have N (D ) = 10, N (T ) = 24 and N (D × Z ) = 54. F 6 F 12 F 6 3 3. The Number of Fuzzy Subgroups of the Group U 6n The groups U were presented for the first time in the famous book of James and 6n Liebeck [22], having presentation 2n 3 U =a, b | a = b = e, bab = a , 6n where we have U = D .Since U is metacyclic, it is supersolvable. Consequently, all maxi- 6 6 6n p j mal subgroups of U have prime index p. They have one of the forms a , b or ab , where 6n p is a prime divisor of 2n and 1 ≤ j ≤ 3. In particular, the number of all maximal subgroups of U is π(2n) + 3, where π is the number of all prime factors of 2n [23]. 6n Next we will determine the number of distinct fuzzy subgroups of U in some particular 6n m m m 1 2 k cases. Let n = p p ··· p be the decomposition of n as a product of prime factors. In 1 2 the process of determining N (U ), we distinguish four cases. F 6n 3.1. p , p , ... , p = 3 Are Distinct Odd Prime Numbers 1 2 In this case, according to the representation of U , there are k + 4 maximal subgroups as 6n follows such that they are of prime index 2, 3, p , ... , p . 1 k ∼ ∼ ∼ M =a Z ; M =ab Z ; M =ab Z ; = = = 1 2n 2 2n 3 2n 2 2 M =a , b =a b Z (since 3 n); 4 3n G =a , b U , where 1 ≤ i ≤ k. i 6n/p The intersection of arbitrary maximal subgroups is as follows: M ∩ M = M ∩ M ∩ M = M ∩ M ∩ M ∩ M =a Z ; u v u v w 1 2 3 4 n p ···p i i r ∼ M ∩ G ∩ ··· ∩ G =a Z ; 1 i i 2n/p ···p 1 r i i p ···p i i ∼ M ∩ G ∩ ··· ∩ G =a b = Z ; 2 i i 2n/p ···p 1 r i ir p ···p 2 i i r ∼ M ∩ G ∩ ··· ∩ G =a b Z ; 3 i i 2n/p ···p 1 r i i 2p ···p 2p ···p i i i i ∼ r r 1 1 M ∩ G ∩ ··· ∩ G =a , b =a b = Z ; 4 i i 3n/p ···p 1 r i ir 2p ···p i i r ∼ M ∩ M ∩ G ∩ ··· ∩ G =a Z ; u v i i n/p ···p 1 r i i 2p ···p i i ∼ M ∩ M ∩ M ∩ G ∩ ··· ∩ G =a = Z ; u v w i i n/p ···p 1 r i i 2p ···p i i r ∼ M ∩ M ∩ M ∩ M ∩ G ∩ ··· ∩ G =a Z ; 1 2 3 4 i i n/p ···p 1 r i i p ···p i i ∼ G ∩ G ∩ ··· ∩ G =a , b = U ; i i i 6n/p ···p 1 2 r i i 1 FUZZY INFORMATION AND ENGINEERING 5 where 1 ≤ u, v, w ≤ 4, u
= v
= w and 1 ≤ i ≤ k are distinct elements for all 1 ≤ j ≤ r. Therefore, by Theorem 2.2, we obtain N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) F 6n F 2n F 3n F n k k + (−1) 3N (Z ) + N (Z ) F 2n/p ···p F 3n/p ···p i i i i 1 r 1 r r=1 i ,i ,...,i =1 1 2 r i <i <···<i 1 2 r k k − (−1) 3N (Z ) + N (U ) .(2) F n/p ···p F 6n/p ···p i i i i r r ⎠ 1 1 r=1 i ,i ,...,i =1 1 2 r i <i <···<i 1 2 r 1 m If k = 1, i.e. n = p = p , then by Equation (2) and Corollary 2.1, we obtain m m m m N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) − 3N (Z m−1 ) − N (Z m−1 ) F 6p F 2p F 3p F p F F 2p 3p +3N (Z m−1 ) + N (U m−1 ) F F p 6p m+2 m = 2N (U m−1 ) + 2 m + 2 · 9. 6p Therefore, by solving the above recurrence relation, we get m 2 N (U m ) = 2 2m + 11m + 10.(3) F 6p In particular, N (U ) = 46. F 6p m m 1 2 If k = 2, i.e. n = p p , then Equation (2) leads to 1 2 N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) F 6n F 2n F 3n F n + −3N (Z ) − N (Z ) + 3N (Z ) + N (U ) F 2n/p F 3n/p F n/p F 6n/p i i i i 1 1 1 1 i =1 + 3N (Z ) + N (Z ) − 3N (Z ) − N (U ) . F 2n/p p F 3n/p p F n/p p F 6n/p p i i i i i i i i 1 2 1 2 1 2 1 2 ⎠ i ,i =1 1 2 i <i 1 2 6 L. KAMALI ARDEKANI AND B. DAVVAZ Consequently, we obtain m m N (U ) F 1 2 6p p 1 2 m m m m m m = 2 3N (Z ) + N (Z ) − 3N (Z ) F 1 2 F 1 2 F 1 2 2p p 3p p p p 1 2 1 2 1 2 − 3N (Z ) − N (Z ) + 3N (Z ) + N (U ) m −1 m m −1 m m −1 m m −1 m F 1 2 F 1 2 F 1 2 F 1 2 2p p 3p p p p 6p p 1 2 1 2 1 2 1 2 − 3N (Z m m −1 ) − N (Z m m −1 ) + 3N (Z m m −1 ) + N (U m m −1 ) F F F F 1 2 1 2 1 2 1 2 2p p 3p p p p 6p p 1 2 1 2 1 2 1 2 + 3N (Z m −1 m −1 ) + N (Z m −1 m −1 ) − 3N (Z m −1 m −1 ) F F F 1 2 1 2 1 2 2p p 3p p p p 1 2 1 2 1 2 −N (U m −1 m −1 ).(4) 1 2 6p p 1 2 For instance, by Equation (4) and Corollary 2.1, we have N (U ) = 2 4N (Z ) − 11N (Z ) + 10N (Z ) + 2N (U ) − N (D ) − 3 F 6p p F 2p p F p p F p F 6p F 6 1 2 1 2 1 2 1 1 = 274. (5) If m = m = ··· = m , then we have 1 2 k ∼ ∼ U = U and Z = Z , 6n/p ···p 6n/p ···p cn/p ···p cn/p ···p i i 1 r i i 1 r r r 1 1 where c = 1, 2, 3. Therefore, by Equation (2), we obtain N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) F 6p ···p F 2p ···p F 3p ···p F p ···p 1 1 1 1 k k k k r−1 + (−1) N (U ) F 6n/p ···p 1 r r=1 + (−1) 3N (Z ) + N (Z ) − 3N (Z ) . F 2n/p ···p F 3n/p ···p F n/p ···p 1 r 1 r 1 r r=1 By Theorem 2.1, N (Z ) = N (Z ) and N (Z ) = N (Z ). There- F 2p ···p F 3p ···p F 2n/p ···p F 3n/p ···p 1 1 1 r 1 r k k fore, we get the following theorem. Theorem 3.1: Let p , ... , p
= 3 be distinct odd prime numbers. The number of all distinct 1 k fuzzy subgroups of the group U is given by the equality 6p ···p 1 k N (U ) = 2 4N (Z ) − 3N (Z ) F 6p ···p F 2p ···p F p ···p 1 k 1 k 1 k + (−1) 4N (Z ) − 3N (Z ) − N (U ) . F 2n/p ···p F n/p ···p F 6n/p ···p 1 r 1 r 1 r r=1 FUZZY INFORMATION AND ENGINEERING 7 For example, by Equations (5), Theorems 2.1 and 3.1, we have N (U ) = 2 4N (Z ) − 15N (Z ) + 21N (Z ) + 3N (U ) F 6p p p F 2p p p F p p p F p p F 6p p 1 2 3 1 2 3 1 2 3 2 3 2 3 −13N (Z ) − 3N (U ) + N (D ) + 3 F p F 6p F 6 3 3 = 2014. 3.2. p = 2 and p , ... , p = 3 Are Distinct Odd Prime Numbers 1 2 k In this case, according to the representation of U , there are k + 3 maximal subgroups of 6n prime index 3, p , ... , p . They are M and G , where 1 ≤ i ≤ 4and 2 ≤ j ≤ k, defined as in 1 k i j Section 3.1. By Theorem 2.2, we obtain N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) F 6n F 2n F 3n F n k−1 k + (−1) 3N (Z ) + N (Z ) F 2n/p ···p F 3n/p ···p i ir i ir 1 1 r=1 i ,i ,...,i =2 1 2 r i <i <···<i 1 2 r k−1 k − (−1) 3N (Z ) + N (U ) ,(6) F n/p ···p F 6n/p ···p i i i i 1 r 1 r ⎠ r=1 i ,i ,...,i =2 1 2 r i <i <···<i 1 2 r where the above-iterated sums are equal to 0 for k = 1. 1 m Suppose that k = 1, i.e. n = p = 2 , then for m = 1, the group U is isomorphic to T and so N (U ) = 24. If m ≥ 2, then by Equation (6) and Corollary 2.1, we obtain 12 F 12 m+1 m m m N (U ) = 2 3N (Z m+1 ) + N (Z ) − 3N (Z ) = 2 (5 + m). F 6·2 F F 3·2 F 2 m 2 If k = 2, i.e. n = 2 p , then Equation (6) leads to N (U ) = 2 (3N (Z ) + N (Z ) − 3N (Z ) F 6n F 2n F 3n F n + −3N (Z ) − N (Z ) + 3N (Z ) + N (U ) F 2n/p F 3n/p F n/p F 6n/p i i i i 1 1 1 1 i =2 = 2 3N (Z m ) + N (Z m ) − 3N (Z m ) − 3N (Z ) 2 2 2 m −1 F m +1 F m F m F 2 1 1 1 m +1 2 p 3·2 p 2 p 1 2 p 2 2 2 −N (Z ) + 3N (Z ) + N (U ).(7) m −1 m −1 m −1 F m 2 F m 2 F m 2 1 1 1 3·2 p 2 p 6·2 p 2 2 2 For example, suppose that p
= 3 is an odd prime number. Then, by Equation (7) and Theorem 2.1, we have N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) F 12p F 4p F 6p F 2p − 3N (Z ) − N (Z ) + 3N (Z ) + N (U )) F 4 F 6 F 2 F 12 = 136. (8) 8 L. KAMALI ARDEKANI AND B. DAVVAZ ∼ ∼ If m = ··· = m , then we have U U and Z Z , = = 2 k 6n/p ···p 6n/p ···p cn/p ···p cn/p ···p i i 2 r+1 i i 2 r+1 r r 1 1 where c = 1, 2, 3. Therefore, Equation (6) leads to the following theorem. Theorem 3.2: Let p = 2,p , ... , p
= 3 be distinct odd prime numbers. The number of all 1 2 distinct fuzzy subgroups of the group U is given by the following equality 12p ···p N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) F 12p ···p F 4p ···p F 6p ···p F 2p ···p 2 k 2 k 2 k 2 k k−1 k − 1 + (−1) 3N (Z ) + N (Z ) F 2n/p ···p F 3n/p ···p 2 r+1 2 r+1 r=1 k−1 k − 1 .(9) − (−1) 3N (Z ) + N (U ) F n/p ···p F 6n/p ···p 2 r+1 2 r+1 r=1 For example, by Equations (8), (9), Theorems 2.1 and 3.2, we have N (U ) = 2 3N (Z ) + N (Z ) − 3N (Z ) − 6N (Z ) − 2N (Z ) F 12p p F 4p p F 6p p F 2p p F 4p F 6p 2 3 2 3 2 3 2 3 3 3 +6N (Z ) + 2N (U ) + 3N (Z ) + N (Z ) − 3N (Z ) − N (U ) F 2p F 12p F 4 F 6 F 2 F 12 3 3 = 968. 3.3. p , p , ... , p Are Distinct Odd Prime Numbers Such that p = 3 1 2 k 1 In this case, there are k + 4 maximal subgroups M and G , where 1 ≤ i ≤ 4and 1 ≤ j ≤ k,of i j prime index 2, p , p , ... , p . The subgroup M =a , b is isomorphic to Z × Z and other 1 2 k 4 n 3 maximal subgroups are defined as in Section 3.1. In the process of determining N (U ), F 6n m m m 1 2 k where n = p p ··· p and p = 3, we distinguish some cases. 1 2 Case 1: n = 3 ,i.e. k = 1. For m = 1, the group U is isomorphic to D × Z . It follows that N (U ) = N (D × 18 6 3 F 18 F 6 Z ) = 54, 7 [Theorem 3.1]. Assume that m ≥ 2, then according to the representation of U , there are 5 maximal subgroups of prime index 2 or 3 that are defined above. The 6·3 intersection of arbitrary maximal subgroups is as follows: M ∩ M = M ∩ M ∩ M = M ∩ M ∩ M ∩ M =a = Z , u v u v w 1 2 3 4 3 3 i−1 6 ∼ ∼ M ∩ G =a b = Z m−1, M ∩ G =a , b = Z m−1 × Z ; i 4 3 2·3 3 where 1 ≤ u, v, w ≤ 4, u
= v
= w and 1 ≤ i ≤ 3. Other intersections of arbitrary maximal subgroups are a Z m−1. Therefore, by Theorem 2.2 and Corollary 2.1, we get m m m N (U ) = 2 3N (Z ) + N (Z × Z ) + N (U m−1 ) F 6·3 F 2·3 F 3 3 F 6·3 − 3N (Z m−1 ) − N (Z m−1 × Z ) F F 3 2·3 3 −3N (Z ) + 3N (Z m−1 ) F 3 F m m−1 = 2 N (U m−1 ) + N (Z × Z ) − N (Z m−1 × Z ) + 3 · 2 + 3 · 2 m . F F 3 3 F 3 6·3 3 The above recurrence relation permits us to determine N (U ) as follows: F 6·3 FUZZY INFORMATION AND ENGINEERING 9 Theorem 3.3: The number of all distinct fuzzy subgroups of the group U m, where m ≥ 2,is 6·3 given by the following equality m−1 m−1 2 m m N (U ) = 2N (Z × Z ) − 2 N (Z × Z ) + 3 × 2 m + 5m + 12 F 6·3 F 3 3 F 3 3 m−2 + 2 N (Z × Z ), m−i F 3 i=1 where the above-iterated sum is equal to 0 for m = 2. For example, by Theorems 2.3 and 3.3, we have N (U ) = 2N (Z × Z ) − 2N (Z × Z ) + 156 = 200. F 54 F 9 3 F 3 3 m m m 2 k Case 2: n = 3 p ··· p such that m = m = ··· = m = 1. 1 2 k By structure of maximal subgroups of U , intersection of them is as follows: 18p ···p 2 k M ∩ G =a b = Z 4 1 n 6p ···p i ir ∼ M ∩ G ∩ G ∩ ··· ∩ G =a b = Z ; 4 1 i i n/p ···p 1 r i i 2p ···p i i ∼ M ∩ G ∩ ··· ∩ G =a , b = Z × Z ; 4 i i n/p ···p 3 1 r i i 1 r where G , ... , G
= G and 1 ≤ r ≤ k. Other intersection of arbitrary maximal subgroups i i 1 1 r are as in Section 3.1. Similarly, by Theorem 2.2, we obtain the following theorem. Theorem 3.4: Let n = p p , ... , p be the decomposition of n as a product of prime factors, 1 2 where p = 3 and p , ... , p
= 3 are distinct odd prime numbers. The number of all distinct 1 2 fuzzy subgroups of the group U is given by the equality 18p ···p N (U ) = 2 3N (Z ) + N (Z × Z ) − 4N (Z ) + N (U ) F 18p ···p F 6p ···p F 3p ···p 3 F 3p ···p F 6p ···p 2 k 2 k 2 k 2 k 2 k + 3 (−1) N (Z ) − N (Z ) F 2n/p ···p F n/p ···p 1 r 1 r r=1 k−1 k − 1 + (−1) N (Z × Z ) + N (U ) F n/p ···p 3 F 6n/3p ···p 2 r+1 2 r+1 r=1 k−1 k − 1 − (−1) N (Z ) + N (U ) . F n/p ···p F 6n/p ···p 2 r+1 2 r+1 r=1 For example, suppose that p
= 3 is an odd prime number. By Theorems 2.1 and 3.4, we get N (U ) = 2 N (Z × Z ) − N (Z × Z ) + N (U ) + N (U ) − N (U ) F 18p F 3p 3 F 3 3 F 6p F 18 F 6 +3N (Z ) − 10N (Z ) + 10N (Z ) − 3 F 6p F 2p F p = 230 + 2N (Z × Z ). F 3p 3 By Theorem 2.2, one can give N (Z × Z ) = 46 which leads to N (U ) = 322. F 3p 3 F 18p 10 L. KAMALI ARDEKANI AND B. DAVVAZ 3.4. p , p , ... , p Are Distinct Prime Numbers Such that p = 2 and p = 3 1 2 1 2 In this case, there are k + 3 maximal subgroups of types M and G defined in Section 3.3, i j where 1 ≤ i ≤ 4and 2 ≤ j ≤ k. Similar to Case 2 in Section 3.3, we get the following theorem. Theorem 3.5: Let n = p p ··· p be the decomposition of n as a product of prime factors, 1 2 k where p = 2,p = 3 and p , ... , p
= 3 are distinct odd prime numbers. The number of all 1 2 3 k distinct fuzzy subgroups of the group U is given by the following equality 36p ···p N (U ) = 2 3N (Z ) + N (Z × Z ) − 4N (Z ) + N (U ) F 36p ···p F 12p ···p F 6p ···p 3 F 6p ···p F 12p ···p 3 k 3 k 3 k 3 k 3 k k−1 k − 1 + 3 (−1) N (Z ) − N (Z ) F 2n/p ···p F n/p ···p 2 r+1 2 r+1 r=1 k−2 k − 2 + (−1) N (Z × Z ) − N (Z ) F n/p ···p 3 F n/p ···p 3 r+2 3 r+2 r=1 k−2 k − 2 + (−1) N (U ) − N (U ) , F 6n/3p ···p F 6n/p ···p 3 r+2 3 r+2 r=1 where the above-iterated sums are equal to 0 for k = 2. For instance, since N (Z × Z ) = 46 7 [Section 4], then by Corollary 2.1 and F 6 3 Theorem 3.5, we conclude that N (U ) = 2 (3N (Z ) + N (Z × Z ) − 4N (Z ) + N (U ) F 36 F 12 F 6 3 F 6 F 12 − 3N (Z ) + 3N (Z ) = 176. F 4 F 2 4. The Number of Fuzzy Normal Subgroups of the Group U 6n 2n ∼ p ∼ Every minimal normal subgroups of U has one of the forms b Z or a Z , where = = 6n 3 p p is a prime divisor of n. In particular, the number of all minimal normal subgroups of U is 6n π(n) + 1. Furthermore, every maximal normal subgroups of U has the form a , b , where 6n p is a prime divisor of 2n, and the number of all maximal normal subgroups of U is π(2n), 6n for more details, see [23]. m m m 1 2 k Let n = p p ··· p be the decomposition of n as a product of distinct prime factors. 1 2 In next, we will determineN (U ), the number of fuzzy normal subgroups of U , in some FN 6n 6n particular cases n = p and n = p p . In order to do this, we calculate the number of all 1 2 chains of normal subgroups of U that terminate in U . If we denote by L , the set of 6n 6n n all chains of length n consisting of normal subgroups of U excluding trivial one element 6n subgroup terminating in U , then N (U ) = 2 + 2 |L |. 6n FN 6n n n∈N n≥2 FUZZY INFORMATION AND ENGINEERING 11 4.1. Case 1:n = p m−1 2p ∼ ∼ In this case, minimal normal subgroups are m =b Z and m =a Z .For = = 0 3 1 p determining N (U m ), we distinguish two cases, p is an odd prime number and p is an FN 6p even prime number. 4.1.1. p Is an Odd Prime. 2 p If p is an odd prime, then maximal normal subgroups are M =a , b and M =a , b 0 1 ∼ ∼ U .If p = 3, then M Z m × Z and otherwise M =a b Z m. m−1 = = 0 3 3 0 3p 6p Assume that m = 1. If p = 3, then U D × Z and by 7 [Theorem 3.2], N (U ) = 18 6 3 FN 18 16. If p
= 3, all non-trivial normal subgroups of U are M and m ,0 ≤ i ≤ 1. Maximal chains 6p i i of normal subgroups of U are one of the following types: 6p {e}⊂ m ⊂ M ⊂ U , {e}⊂ m ⊂ M ⊂ U , 0 i 6p 1 0 6p where i = 0, 1. Thus, |L |= 4and |L |= 3 which are described as follows: 2 3 M ⊂ U , m ⊂ U , m ⊂ M ⊂ U , m ⊂ M ⊂ U , i 6p i 6p 0 i 6p 1 0 6p where i = 0, 1. Hence, we get N (U ) = 16. FN 6p If m = 2, then all non-trivial normal subgroups of U are N =a Z , N = 2 = 2 1 2 6p p 2p p ∼ ∼ a , b , N =a , b = D , M and m ,0 ≤ i ≤ 1. If p = 3, then N = Z × Z and oth- 3 6 i i 2 3 3 2p erwise N =a b = Z . Maximal chains of normal subgroups of U 2 are one of the 2 3p 6p following types: {e}⊂ m ⊂ N ⊂ M ⊂ U 2, {e}⊂ m ⊂ N ⊂ M ⊂ U 2, 0 2 i 1 2 i 6p 6p {e}⊂ m ⊂ N ⊂ M ⊂ U 2, {e}⊂ m ⊂ N ⊂ M ⊂ U 2, 0 3 1 1 1 0 6p 6p where i = 0, 1. Therefore, we conclude that |L |= 7, |L |= 12 and |L |= 6which are 2 3 4 described as follows: M ⊂ U 2, m ⊂ U 2, N ⊂ U 2, i i j 6p 6p 6p m ⊂ M ⊂ U 2, N ⊂ M ⊂ U 2, m ⊂ M ⊂ U 2, i 0 i+1 0 i 1 6p 6p 6p N ⊂ M ⊂ U 2, m ⊂ N ⊂ U 2, m ⊂ N ⊂ U 2, i+2 1 0 i+2 1 i+1 6p 6p 6p m ⊂ N ⊂ M ⊂ U 2, m ⊂ N ⊂ M ⊂ U 2, m ⊂ N ⊂ M ⊂ U 2, 0 2 i 1 2 i 0 3 1 6p 6p 6p m ⊂ N ⊂ M ⊂ U 2, 1 1 0 6p where i = 0, 1 and 1 ≤ j ≤ 3. Therefore, we deduce that N (U 2 ) = 52. FN 6p If m = 3, then all non-trivial normal subgroups of U 3 are N =a = Z 3, N = 1 2 6p p 3 2 2 2p p 2p p 2p ∼ ∼ ∼ a , b , N =a , b D , N =a Z 2, N =a , b U , N =a , b , M and = = = 3 6 4 5 6p 6 i 2p ∼ ∼ ∼ m ,0 ≤ i ≤ 1. If p = 3, then N Z × Z and N Z × Z . Otherwise N =a b = = = i 2 9 3 6 3 3 2 2p Z 2 and N =a b Z . Maximal chains of normal subgroups of U 3 are one of the 3p 6 3p 6p following types: {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U 3, {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U 3, i 6 2 0 i 6 2 1 6p 6p {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U 3, {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U 3, i 6 5 1 0 3 5 1 6p 6p {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U 3, {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U 3, 1 4 2 i 1 4 1 0 6p 6p where i = 0, 1. Consequently, |L |= 10, |L |= 27, |L |= 28 and |L |= 10 which leads 2 3 4 5 to N (U 3 ) = 152. FN 6p 12 L. KAMALI ARDEKANI AND B. DAVVAZ 4.1.2. p Is an Even Prime. 2 2 Suppose that p = 2. Then, maximal normal subgroup is M =a , b =a b Z m. 0 3·2 If m = 1, then U T and soN (U ) = 12, by 7 [Theorem 3.2]. Assume that m = 2, 12 12 FN 12 2 4 4 ∼ ∼ then all normal subgroups of U are N =a Z , N =a , b =a b Z , M and = = 24 1 4 2 6 1 m , i = 0, 1. Maximal chains of normal subgroups of U are one of the following types: i 24 {e}⊂ m ⊂ N ⊂ M ⊂ U , {e}⊂ m ⊂ N ⊂ M ⊂ U , i 2 0 24 1 1 0 24 where i = 0, 1. Consequently, |L |= 5, |L |= 7and |L |= 3 which are described as 2 3 4 follows: M ⊂ U , m ⊂ U , N ⊂ U , 0 24 i 24 i+1 24 m ⊂ M ⊂ U , m ⊂ N ⊂ U , N ⊂ M ⊂ U , i 0 24 i 2 24 i+1 0 24 m ⊂ N ⊂ U , m ⊂ N ⊂ M ⊂ U , m ⊂ N ⊂ M ⊂ U , 1 1 24 i 2 0 24 1 1 0 24 where i = 0, 1. Consequently, N (U ) = 32. FN 24 2 4 If m = 3, then all non-trivial normal subgroups of U are N =a = Z , N =a , b = 48 1 8 2 4 8 8 4 ∼ ∼ ∼ a b = Z , N =a , b =a b = Z , N =a = Z , M and m , i = 0, 1. Maximal 12 3 6 4 4 0 i chains of normal subgroups of U are one of the following types: {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U , {e}⊂ m ⊂ N ⊂ N ⊂ M ⊂ U , i 3 2 0 48 1 4 i+1 0 48 where i = 0, 1. Consequently, |L |= 7, |L |= 15, |L |= 13 and |L |= 4whichleadsto 2 3 4 5 N (U ) = 80. FN 48 4.2. Case 2:n = p p 1 2 Suppose that p and p are distinct odd prime numbers such that p
= 3. In this case, 1 2 2 2p 2p ∼ 2 ∼ 1 ∼ minimal normal subgroups are m =b = Z , m =a = Z and m =a = Z . 0 3 1 p 2 p 1 2 2 p p 1 ∼ 2 ∼ Maximal normal subgroups are M =a , b , M =a , b = U and M =a , b = 0 1 6p 2 ∼ ∼ U .If p = 3, then M = Z × Z and otherwise M =a b = Z . 6p 1 0 3p 3 0 3p p 1 2 1 2 2p 2p 1 1 ∼ All non-trivial normal subgroups of U are N =a , b =a b = Z , N = 6p p 1 3p 2 1 2 2 2p p p 2 2 1 2 ∼ ∼ ∼ a , b , N =a , b = D and N =a = Z .If p = 3, then N = Z × Z and 3 6 4 p p 1 2 3 3 1 2 2p 2 ∼ otherwise N =a b = Z . Maximal chains of normal subgroups of U are one of 2 3p 6p p 1 1 2 the following types: {e}⊂ m ⊆ N ⊆ M ⊆ U , {e}⊂ m ⊆ N ⊆ M ⊆ U , i 2 0 6p p i 2 2 6p p 1 2 1 2 {e}⊂ m ⊆ N ⊆ M ⊆ U , {e}⊂ m ⊆ N ⊆ M ⊆ U , j 1 0 6p p j 1 1 6p p 1 2 1 2 {e}⊂ m ⊆ N ⊆ M ⊆ U , {e}⊂ m ⊆ N ⊆ M ⊆ U , 0 3 k 6p p k 4 0 6p p 1 2 1 2 where i = 0, 1; j = 0, 2; k = 1, 2. Consequently, |L |= 10, |L |= 21 and |L |= 12 which 2 3 4 leads to N (U ) = 88. FN 6p p 1 2 Suppose that p = 2and p is an odd prime number. In this case, minimal normal sub- 1 2 2p 4 ∼ 2 ∼ ∼ groups are m =b Z , m =a Z and m =a Z . Maximal normal sub- = = = 0 3 1 2 2 p 2 p 2 ∼ ∼ groups are M =a , b , M =a , b U .If p = 3, then M Z × Z and otherwise = = 0 1 12 2 0 6 3 M =a b Z . 0 6p 4 2p 2p 2 2 ∼ All non-trivial normal subgroups of U are N =a , b , N =a , b =a b 12p 1 2 2 4 ∼ ∼ ∼ Z and N =a Z .If p = 3, then N Z × Z and otherwise N =a b Z . = = = 6 3 2p 2 1 3 3 1 3p 2 2 FUZZY INFORMATION AND ENGINEERING 13 Table 1. The exact number of distinct fuzzy subgroups of some groups U . 6n Group N (U ),(by Section 3) F 6n U 54 U 176 U 200 m+1 U 2 (5 + m) 6·2 m 2 U 2 (2m + 11m + 10), p
= 3 is an odd prime number 6p U 136, p
= 3 is an odd prime number 12p U 322, p
= 3 is an odd prime number 18p U 274, p , p
= 3 are distinct odd prime numbers 6p p 1 2 1 2 U 968, p , p
= 3 are distinct odd prime numbers 12p p 2 3 2 3 U 2014, p , p , p
= 3 are distinct odd prime numbers 6p p p 1 2 3 1 2 3 Table 2. The exact number of distinct fuzzy normal subgroups of some groups U . 6n Group N (U ),(bySection 4) FN 6n U 12 U 32 U 80 U 16, p is an odd prime number 6p U 2 52, p is an odd prime number 6p U 3 152, p is an odd prime number 6p U 64, p is an odd prime number 12p 2 U 88, p , p are distinct odd prime numbers 6p p 1 2 1 2 Maximal chains of normal subgroups of U are one of the following types: 12p {e}⊂ m ⊆ N ⊆ M ⊆ U , {e}⊂ m ⊆ N ⊆ M ⊆ U , i 2 0 12p i 2 1 12p 2 2 {e}⊂ m ⊆ N ⊆ M ⊆ U , {e}⊂ m ⊆ N ⊆ M ⊆ U , j 1 0 12p k 3 0 12p 2 2 where i = 0, 1; j = 0, 2; k = 1, 2. Consequently, |L |= 8, |L |= 15 and |L |= 8which 2 3 4 leads to N (U ) = 64. FN 12p 5. Conclusion In this paper, we have been solving the problem of the number of distinct fuzzy subgroups of finite groups U relative to the natural equivalence relation ∼ . Furthermore, we studied 6n the subgroup structure of U and gave a recurrence relation for determining F(U ).By 6n 6n solving obtained recurrence relation, we found explicit formulas for the number of distinct fuzzy subgroups of U m, U m m and U . For some special cases of groups U , 1 2 6p 6p p 6p p ···p 6n 1 2 1 2 k the exact number of distinct fuzzy subgroups and fuzzy normal subgroups with respect to ∼ is given in Tables 1 and 2, respectively. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributors Leili Kamali Ardekani is assistant professor at the Faculty of Engineering, Ardakan University, Iran. She received her M.Sc. in 2010 and Ph.D in 2014 on Pure Mathematics (Algebra), both from Yazd University, 14 L. KAMALI ARDEKANI AND B. DAVVAZ Iran. Since 2015, she has been a faculty member at the Faculty of Engineering in Ardakan University, Ardakan, Iran. Her research interests are in the areas of algebraic hyperstructure theory, fuzzy logic group representation theory and finite groups. Bijan Davvaz is Professor at the Department of Mathematics, Yazd University, Iran. He earned his Ph.D. in Mathematics from Tarbiat Modarres University, Iran. His areas of interest include algebra, algebraic hyperstructures, rough sets and fuzzy logic. He is a member of editorial boards for 25 mathematical journals. Prof. Davvaz has authored 6 books and over 600 research papers, especially on algebra, fuzzy logic, algebraic hyperstructures and their applications. ORCID L. Kamali Ardekani http://orcid.org/0000-0002-9942-6356 References [1] Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–353. [2] Rosenfeld A. Fuzzy groups. J Math Anal Appl. 1971;35:512–517. [3] Wu WM. Normal fuzzy subgroups. Fuzzy Math. 1981;1:21–30. [4] Mukherjee NP, Bhattacharya P. Fuzzy normal subgroups and fuzzy cosets. Inform Sci. 1984;34(3):225–239. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Sep 21, 2022
Keywords: Equivalence relation; lattice; fuzzy subgroup; chain of subgroups; level subset; metacyclic group; 20N25; 20E15