Constructing Fuzzy Subgroups of Symmetric Groups S 4

Transcription

1 International Journal of Algebra, Vol 6, 2012, no 1, Constructing Fuzzy Subgroups of Symmetric Groups S 4 R Sulaiman Department of Mathematics, Faculty of Mathematics and Sciences Universitas Negeri Surabaya, Surabaya Indonesia sulaimanraden@yahoocom Abstract This article computes the number of fuzzy subgroups of symmetric group S 4 and constructs some of them First, an equivalence relation on the set of all fuzzy subgroups of a group G is defined The diagram of subgroups lattice of S 4 is used to determine the number of fuzzy subgroups of S 4 We find the total number of fuzzy subgroups of S 4 is 220 Mathematics Subject Classification: 20B30, 20B35, 03G10 Keywords: Equivalence, fuzzy subgroup, permutation group 1 Introduction After the first paper by Zadeh in 1965 [3], several aspects of fuzzy subset were studied in the first fifteen years The study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy subgroups by Rosenfeld in 1971 [1] Without any equivalence relation on fuzzy subgroups of group G, the number of fuzzy subgroups is infinite, even for the trivial group {e} Some authors have used the equivalence relation of fuzzy sets to study the equivalence of fuzzy subgroups ([2], [4], [5], [9], [10], [11]) All of them have treated the particular case of finite abelian group It is interesting to count the number of fuzzy subgroups of nonabelian groups and construct them Laszlo in [2] has studied about the construction of fuzzy subgroup of group of order one to six In the first paper of this topic, Sulaiman and Abd Ghafur [6] have counted the number of fuzzy subgroups of nonabelian symmetric groups S 2,S 3 and alternating group A 4 In the other paper, they [7] have counted the number of fuzzy subgroups of group defined by a presentation

2 24 R Sulaiman In [8] we have constructed the diagram of subgroups lattice of symmetric group S 4 That result is very important for this paper, in order to count the number of fuzzy subgroup of S 4 2 Preliminary Notes Definition 21 Let G be a group A function μ from G into [0,1] is called a fuzzy subgroup of G if μ(xy) min{μ(x),μ(y)}, x, y G and μ(x 1 ) μ(x), x G Theorem 22 Sulaiman and Abd Gahfur [6] Function μ : G [0, 1] is a fuzzy subgroup of G if there is a chain P 1 <P 2 < < P n = G in subgroups lattice of G such that μ can be written as μ(x) = θ 1, x P 1 θ 2, x P 2 \ P 1 θ n, x P n \ P n 1 Without any equivalence relations on fuzzy subgroups of a group G, the number of fuzzy subgroups is infinite even for the trivial group {e} So we define equivalence relation on the set of all fuzzy subgroups of a given group Using this equivalent, we have constructed all of the fuzzy subgroups of symmetric groups S 4 (1) Definition 23 Let μ, γ be fuzzy subgroups of G of the form μ(x) = θ 1, x P 1 θ 2, x P 2 \ P 1 θ n, x P n \ P n 1, γ(x) = δ 1, x M 1 δ 2, x M 2 \ M 1 δ n, x M m \ M m 1 Then we say that μ and γ are equivalent and write μ γ, if (1)m = n and (2)P i = M i, i {1, 2,, m} It is easily checked that this relation is indeed an equivalence relation Two fuzzy subgroups of G are said to be different if they are not equivalent Example 24 Consider the group G = Z 12 Letμ, γ, α, β be functions from Z 12 into [0, 1] such that { { 1, x {0, 2, 4, 6, 8, 10} 1, x {0, 1, 2, 3, 4, 5} μ(x) = 1, γ(x) = 1, x {1, 3, 5, 7, 9, 11}, x {6, 7, 8, 9, 10, 11} 2 2

3 Fuzzy subgroup , x {0, 4, 8}, x {0, 4, 8} α(x) =, x {2, 6, 10} 2, β(x) =, x {2, 6, 10} 1, x {1, 3, 5, 7, 9, 11} 4 0, x {1, 3, 5, 7, 9, 11} 3 Note that P 1 (μ) ={0, 2, 4, 6, 8, 10},P 2 (μ) =Z 12 Thus P 1 (μ),p 2 (μ) are both subgroups of Z 12 By Theorem 22, μ is a fuzzy subgroup of Z 12 Similarly, we can show that α and β both are fuzzy subgroups of Z 12 Note that P 1 (γ) is not a subgroup of Z 12 Hence γ is not a fuzzy subgroup of Z 12 Since μ α and μ β,by Definition 23 we have μ is not equivalent with α and μ are not equivalent with β Note that α = β and it is easy to show that P i (α) =P i (β), i {1, 2, 3} Thus α β Lemma 25 The number of fuzzy subgroups of G is equal to the number of chain on the subgroups lattice of G Proof: Using Theorem 22 and Definition 23 3 Constructing fuzzy subgroups of S 4 Let S 4 = {i = id, σ 1,σ 2,, σ 9,τ 1,τ 2,, τ 8,α 1,α 2,, α 6 } with σ 1 = (1234),σ 2 = (13)(24), σ 3 = (1432), σ 4 = (1243),σ 5 = (14)(23), σ 6 = (1342), σ 7 = (1324), σ 8 = (12)(34), σ 9 = (1432),τ 1 = (234),τ 2 = (243), τ 3 = (134), τ 4 = (143), τ 5 = (124), τ 6 = (142), τ 7 = (123), τ 8 = (132), α 1 = (12),α 1 = (12),α 2 = (13), α 3 = (14), α 4 = (23), α 5 = (24), α 6 = (34) We have got thirty subgroups of S 4 ( see in [8]) those are, K 1 = {i},k 2 = {i, σ 2 }, K 3 = {i, σ 5 }, K 4 = {i, σ 8 }, K 5 = {i, α 1 }, K 6 = {i, α 2 }, K 7 = {i, α 3 }, K 8 = {i, α 4 }, K 9 = {i, α 5 }, K 10 = {i, α 6 },L 11 = {i, τ 1,τ 2 }, L 12 = {i, τ 3,τ 4 }, L 13 = {i, τ 5,τ 6 }, L 14 = {i, τ 7,τ 8 }, M 15 = {i, σ 1,σ 2,σ 3 },M 16 = {i, σ 4,σ 5,σ 6 }, M 17 = {i, σ 7,σ 8,σ 9 },M 18 = {i, σ 2,σ 5,σ 8 }, M 19 = {i, σ 2,α 2,α 5 }, M 20 = {i, σ 5,α 3,α 4 }, M 21 = {i, σ 8,α 1,α 6 },N 22 = {i, τ 1,τ 2,α 4,α 5,α 6 },N 23 = {i, τ 3,τ 4,α 2,α 3,α 6 },N 24 = {i, τ 5,τ 6,α 1,α 3,α 5 }, N 25 = {i, τ 7,τ 8,α 1,α 2,α 4 }, P 26 = {i, σ 1,σ 2,σ 3,σ 5,σ 8,α 2,α 5 }, P 27 = {i, σ 2,σ 4,σ 5,σ 6,σ 8,α 3,α 4 }, P 28 = {i, σ 2,σ 5,σ 7,σ 8,σ 9,α 1,α 6 }, alternating group A 4 and S 4 itself The diagram of subgroups lattice of S 4 is shown as fig1 We will count the number of the fuzzy subgroups of S 4 by observing that diagram and using Lemma 1 We can see that the maximal chain on that lattice consists of five subgroups of S 4 Therefore, the fuzzy subgroup μ of S 4 has length 1,2,3,4 or 5 Let μ be a fuzzy subgroup of S 4 We will identify μ according to P 1 (μ) Every subgroup of S 4 can be chosen to be P 1 (μ) fig1

4 26 R Sulaiman Figure 1: Subgroup lattice of S 4 If P 1 (μ) =S 4, we only have one fuzzy subgroup of S 4, that is μ 1 (x) = θ 1, x S 4 If P 1 (μ) =A 4, then the only option we have is S 4 = P 2 (μ) Therefore we only have one fuzzy subgroup of S 4 with P 1 (μ) =A 4 that is μ 2 (x) = { θ1, x A 4 θ 2, x S 4 \A 4 Similarly, we have one fuzzy subgroup for P 1 (μ) =P 26,P 1 (μ) =P 27,P 1 (μ) = P 28,P 1 (μ) =N 22,P 1 (μ) =N 23,P 1 (μ) =N 24,P 1 (μ) =N 25 If P 1 (μ) =M 15, then we have two chains, those are M 15 <S 4 and M 15 <P 26 <S 4 Therefore, we get two fuzzy subgroups of S 4 with P 1 = M 15, those are μ 10 (x) = { θ1, x M 15 θ 2, x S 4 \M 15 and μ 11 (x) = θ 1, x M 15 θ 2, x P 26 \M 15 θ 3, x S 4 \P 26 Similarly, we have two fuzzy subgroups for P 1 μ = M 16,P 1 μ = M 17,P 1 μ = M 19,P 1 μ = M 20,P 1 μ = M 21 If P 1 = M 18, then we have five chains, those are M 18 <P 26 <S 4, M 18 <P 27 <S 4,M 18 <P 28 <S 4 and M 18 <A 4 <S 4 Therefore, we get five fuzzy subgroups of S 4 with P 1 = M 18, those are

5 Fuzzy subgroup 27 μ 22 (x) = μ 24 (x) = { θ1, x M 18 θ 2, x S 4 \M 18, μ 23 (x) = θ 1, x M 18 θ 2, x P 27 \M 18, μ 25 (x) = θ 3, x S 4 \P 27 θ 1, x M 18 μ 26 (x) = θ 2, x A 4 \M 18 θ 3, x S 4 \A 4 θ 1, x M 18 θ 2, x P 26 \M 18 θ 3, x S 4 \P 26 θ 1, x M 18 θ 2, x P 28 \M 18 and θ 3, x S 4 \P 28 If P 1 (μ) =L 11, then we have three chains, those are L 11 <S 4, L 11 < A 4 < S 4 and L 11 < N 22 < S 4 Therefore, we get three fuzzy subgroups of S 4 with P 1 (μ) = L 11 and we have the same number for P 1 (μ) = L 12, P 1 (μ) =L 13,P 1 (μ) =L 14 By similar method, we have (1) Twelve fuzzy subgroups of S 4 for P 1 ()μ = K 2,P 1 ()μ = K 3 and P 1 = K 4 (2) Six fuzzy subgroups of S 4 for P 1 (μ) = K 5, P 1 (μ) = K 6, P 1 (μ) = K 7, P 1 (μ) =K 8, P 1 (μ) =K 9 and P 1 (μ) =K 10 (3) For P 1 (μ) ={e} we have 109 subgroups fuzzy Thus, the total number of fuzzy subgroups of S 4 is 1 + (81) + (62) (43) + (312) + (66) = 220 References [1] A Rosenfeld, Fuzzy groups, Journal of Mathematical AnalAndApp, 35 (1971), [2] F Lazlo, Structure and construction of fuzzy subgroup of a group, Fuzzy Set and System, 51 (1992), [3] LA Zadeh, Fuzzy sets, Inform and Control, 8 (1965), [4] M Tarnauceanu, The number of fuzzy subgroups of finite cyclic groups and Delanoy numbers, European Journal of Combinatoric, 30 (2009), [5] M Tarnauceanu and L Bentea, On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems, 159 (2008), [6] R Sulaiman and Abd Ghafur Ahmad, Counting fuzzy subgroups of symmetric groups S 2,S 3 and alternating group A 4, Journal of Quality Measurement and Analysis, 6 (2010), 57-63

6 28 R Sulaiman [7] R Sulaiman and Abd Ghafur Ahmad, The number of fuzzy subgroups of group defined by apresentation, International Journal of Algebra, 5 no8 (2011), [8] R Sulaiman, Subgroups lattice of symmetric group S(4), International Journal of Algebra, submitted (2011) [9] V Murali and BB Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets and Systems, 123 (2001), [10] V Murali and BB Makamba, On an equivalence of fuzzy subgroups II, Fuzzy Sets and Systems, 136 (2003), [11] Y Zhang and K Zou, A note an equivalence on fuzzy subgroups, Fuzzy Sets and Systems, 95 (1992), Received: August, 2011