The Number of Fuzzy Subgroups of Cuboid Group

Transcription

1 International Journal of Algebra, Vol. 9, 2015, no. 12, HIKARI Ltd, The Number of Fuzzy Subgroups of Cuboid Group Raden Sulaiman Department of Mathematics, Faculty of Mathematics and Sciences Universitas Negeri Surabaya, Surabaya Indonesia Copyright c 2015 Raden Sulaiman. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The main goal of this article is to give some explicit formulas for the number of fuzzy subgroups of cuboid group. The first, we define cuboid group and then construct the formula for cuboid group with m 2 1, m 1 2 and finally for m 2 2 in size. Mathematics Subject Classification: 20B30, 20B35, 03G10 Keywords: Fuzzy Subgroup,Lattice, Lattice Method, Cuboid Group 1 Introduction The starting point of our discussion is given by the paper [1], where it is constructed a method to determine the number of fuzzy subgroups of finite groups. The method is called lattice method. By using this method, in my paper before (see [2])we constructed the formula of the number of fuzzy subgroups of rectangle group. In this paper we propose a definition of cuboid group. Using the diagram of the lattice subgroups, the result in [2], [3], and lattice method, we determine the formula of the number of fuzzy subgroups of cuboid group. 2 Preliminary Notes We recall some definitions and results that will be used later. Some definitions, symbols, and proving of some theorems in this section can be seen in [1] and

2 522 Raden Sulaiman [2]. Definition 2.1 Let X be a nonempty set. A fuzzy subset of X is a function from X into [0, 1]. Definition 2.2 (Rosenfeld, see [4] Let G be a group. A fuzzy subset µ of G is called a fuzzy subgroup of G if (1) µ(xy) min{µ(x), µ(y)}, x, y G, (2) µ(x 1 ) µ(x), x G. Theorem 2.3 (Sulaiman and Abdul Ghafur, see [5] A fuzzy subset µ of G is a fuzzy subgroup of G if and only if there is a chain of subgroups G, P 1 (µ) P 2 (µ) P 3 (µ)... P n (µ) = G such that µ is in the form µ(x) = θ 1, x P 1 (µ) θ 2, x P 2 (µ). θ m, x P m (µ) Definition 2.4 (Sulaiman and Priyo Budi, see [2]) Let G be a group and the number of it s subgroups is finite. The diagram of lattice subgroups of G is called rectangle if satisfies these conditions: The subgroups of G can be labeled by Kj i where 1 i m, 1 j s for some m, s Z with K1 1 = G, Ks m = {e} such that: (i) for fixed i, 1 i m, Kk i < Kk 1, i k, 2 k s, (ii) for fixed j, 1 j m, Kj t < Kj t 1, t, 2 t m. Theorem 2.5 (see Theorem 3 (ii) in [2] Let G be a rectangle group with m N, s = 2. We have: i) n(f P1 =K2 m) = 2.n(F P 1 =K m 1 ) + n(f P1 =K m 2 1 ii) n(f P1 =K2 m) = 2m + (m 3)2 m 2 = 2 m 2 (m Main Results In this article, G is assumed as a group and the number of fuzzy subgroups of G is finite. Definition 3.1 Let G be a group. Group G is called cuboid if the subgroups of G can be labeled by K ijk where 1 i m, 1 j n, 1 k s for some m, n, s Z with K 111 = G, K m,n,s = {e} and satisfies these three conditions: (i) for fixed i, j, 1 i m, 1 j n, K ijk < K ij(k 1), k, 2 k s, (ii) for fixed i, k, 1 i m, 1 k s, K ijk < K i(j 1)k, j, 2 j n,. (iii) for fixed j, k, 1 j n, 1 k s, K ijk < K (i 1)jk, i, 2 i m,. For this case the size of the group is m n s. The diagram of lattice subgroups of cuboid group G is shown in Figure 1. (1)

3 The number of fuzzy subgroups of cuboid group 523 Figure 1: Lattice of Rectangle group Theorem 3.2 Let G be a group that satisfied Definition 3.1 with m N, n = s = 2. We have: i) ) = 2.(F P1 =K (m 1)21 ii) n(f P1 =K m12 ) = 2.(F P1 =K (m 1)12 Proof. By Considering diagram of lattice subgroups G (see figure 2), we have n(f P1 =K i21 ) = n(f P1 =K i12 ), i {1, 2,..., m}. (2) i) Using lattice method (see [1]), we get n(f P1 =K i21 ) = i 1 j=1[n(f P1 =K j21 n(f P1 =K j11 )]+n(f P1 =K i11 ), i {2, 3, 4,..., m}. From this we obtain, and ) = m 1 j=1 [n(f P1 =K j21 ) + n(f P1 =K j11 )] + n(f P1 =K m11 ) (3)

4 524 Raden Sulaiman Figure 2: Rectangle group mx2x2 n(f P1 =K (m 1)21 ) = m 2 ) + n(f P1 =K j11 )] + n(f P1 =K (m 1)11 Equation (3) can be written as ) = m 2 n(f P1 =K j11 )]+n(f P1 =K m11 n(f P1 =K (m 1)21 n(f P1 =K (m 1)11 ) = n(f P1 =K (m 1)11 ) + n(f P1 =K m11 ) + n(f P1 =K (m 1)21 ) = 2.n(F P1 =K (m 1)21 ) + n(f P1 =K m11 ) = 2.n(F P1 =K (m 1)21 ii) From (1) we have ) = n(f P1 =K m12 From part i) of this theorem we obtain ) 2.n(F P1 =K (m 1)12 Theorem 3.3 Let G be a group that satisfied Definition 3.1 with m N, n = s = 2. We have, j=1 n(f P1 =K j22 ) + (2m + 1)2 m 1. Proof. By considering the diagram of lattice G (see figure 2) we get ) + n(f P1 =K j21 ) + n(f P1 =K j12 )] + n(f P1 =K m12 ) + n(f P1 =K m11 (4) According to (2), then (4) can be written as n(f P1 =K m12 n(f P1 =K j21 n(f P1 =K j11 )]+2. 2 m 2. By using (3) we have, 2.n(F P1 =K j21 n(f P1 =K j11 )]+2.( m 1 n(f P1 =K j11 )] + n(f P1 =K m11 ) 4.n(F P1 =K j21 3.n(F P1 =K j11 )]+3.2 m 2.

5 The number of fuzzy subgroups of cuboid group n(F P1 =K j21 )]+3.[n(F P1 =K 111 n(f P1 =K 211 n(f P1 =K 311 ) n(f P1 =K (m 1)11 )] m 2. ) + 4.n(F P1 =K j21 )] m 2. ) = m 1 j=1 Now, consider, m 1 ) = 4. m 1 j=1 n(f P1 =K j21 m 1 j=1 [n(f P1 =K j22 ) + 4.n(F P1 =K j21 )] m 1. (5) [n(f P1 =K j21 ) = 4.[n(F P1 =K 121 n(f P1 =K 221 n(f P1 =K n(F P1 =K (m 1)21 )]. According Theorem 2.5 ii), for every i {1, 2,..., (m 1)}, we have n(f P1 =K i21 ) = 2 i + (i 3)2 i 2 = (i + 1)2 i 2. Hence, (6) can be written as, m 1 ) = 4( m.2 m 3 = 4 ( m.2 m ) = mi=1 i(2 i Since m i=1 i(2 i ) = 2 + (m 1)2 m+1 (see [6];176), we conclude that, m 1 j=1 4.n(F P1 =K j21 ) = (2 + (m 1)2m+1 ) = (m 1)2 m. From this, we can write (5) becomes j=1 n(f P1 =K j22 ) + (m 1)2 m m 1 = m 1 j=1 n(f P1 =K j22 ) + (2m + 1)2 m 1. (6) References [1] R. Sulaiman, Fuzzy Subgroups Computation of Finite Group by Using Their Lattice, International Journal of Pure and Applied Mathematics, 78 (2012), no. 4, [2] R. Sulaiman, Priyo Budi Prawoto, The number of fuzzy subgroups of rectangle groups, International Journal of Algebra, 8 (2014), no. 1, [3] R. Sulaiman, Priyo Budi, Computing the number of fuzzy subgroups by expansion method, International Electronic Journal of Pure and Applied Mathematics, 8 (2014), no. 4,

6 526 Raden Sulaiman [4] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), [5] 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), [6] R.P. Grimaldi, Discrete and Combinatorial Mathematics, Addison Wesley, New York, Received: October 5, 2015; Published: December 12, 2015