International Journal of Algebra, Vol. 9, 2015, no. 12, 521 - 526
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2015.5958
The Number of Fuzzy Subgroups
of Cuboid Group
Raden Sulaiman
Department of Mathematics, Faculty of Mathematics and Sciences
Universitas Negeri Surabaya, Surabaya 60231 Indonesia
c 2015 Raden Sulaiman. This article is distributed under the Creative ComCopyright mons 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
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,
P1 (µ) ≤ P2 (µ) ≤ P3 (µ) ≤ ... ≤ Pn (µ) = G such that µ is in the form






θ1 ,
θ2 ,
µ(x) =
..


 .


x ∈ P1 (µ)
x ∈ P2 (µ)
(1)
θm , x ∈ Pm (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 Kji where 1 ≤ i ≤ m, 1 ≤ j ≤ s for some m, s ∈ Z with K11 =
i
G, Ksm = {e} such that: (i) for fixed i, 1 ≤ i ≤ m, Kki < Kk−1
, ∀k, 2 ≤ k ≤ s,
t−1
t
(ii) for fixed j, 1 ≤ j ≤ m, Kj < Kj , ∀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(FP1 =K2m ) = 2.n(FP1 =K m−1 ) + n(FP1 =K1m ).
2
ii) n(FP1 =K2m ) = 2m + (m − 3)2m−2 = 2m−2 (m + 1).
3
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 Kijk where 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ k ≤ s
for some m, n, s ∈ Z with K111 = G, Km,n,s = {e} and satisfies these three
conditions:
(i) for fixed i, j, 1 ≤ i ≤ m, 1 ≤ j ≤ n, Kijk < Kij(k−1) , ∀k, 2 ≤ k ≤ s,
(ii) for fixed i, k, 1 ≤ i ≤ m, 1 ≤ k ≤ s, Kijk < Ki(j−1)k , ∀j, 2 ≤ j ≤ n,.
(iii) for fixed j, k, 1 ≤ j ≤ n, 1 ≤ k ≤ s, Kijk < 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.
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) n(FP1 =Km21 ) = 2.(FP1 =K(m−1)21 ) + 2m−2 .
ii) n(FP1 =Km12 ) = 2.(FP1 =K(m−1)12 ) + 2m−2 .
Proof. By Considering diagram of lattice subgroups G (see figure 2), we
have
n(FP1 =Ki21 ) = n(FP1 =Ki12 ), ∀i ∈ {1, 2, ..., m}.
(2)
i) Using lattice method (see [1]), we get
n(FP1 =Ki21 ) = i−1
j=1 [n(FP1 =Kj21 )+n(FP1 =Kj11 )]+n(FP1 =Ki11 ), i ∈ {2, 3, 4, ..., m}.
From this we obtain,
P
n(FP1 =Km21 ) =
m−1
X
[n(FP1 =Kj21 ) + n(FP1 =Kj11 )] + n(FP1 =Km11 )
j=1
and
(3)
524
Raden Sulaiman
Figure 2: Rectangle group mx2x2
n(FP1 =K(m−1)21 ) = m−2
j=1 [n(FP1 =Kj21 ) + n(FP1 =Kj11 )] + n(FP1 =K(m−1)11 ).
Equation (3) can be written as
P
n(FP1 =Km21 ) = m−2
j=1 [n(FP1 =Kj21 )+n(FP1 =Kj11 )]+n(FP1 =Km11 )+n(FP1 =K(m−1)21 )+
n(FP1 =K(m−1)11 ).
n(FP1 =Km21 ) = n(FP1 =K(m−1)11 ) + n(FP1 =Km11 ) + n(FP1 =K(m−1)21 ).
n(FP1 =Km21 ) = 2.n(FP1 =K(m−1)21 ) + n(FP1 =Km11 ).
n(FP1 =Km21 ) = 2.n(FP1 =K(m−1)21 ) + 2m−2 .
P
ii) From (1) we have n(FP1 =Km21 ) = n(FP1 =Km12 ). From part i) of this theorem we obtain n(FP1 =Km21 ) − 2.n(FP1 =K(m−1)12 ) + 2m−2 .
Theorem 3.3 Let G be a group that satisfied Definition 3.1 with m ∈
N, n = s = 2. We have,
P
m−1
n(FP1 =Km22 ) = m−1
.
j=1 n(FP1 =Kj22 ) + (2m + 1)2
Proof. By considering the diagram of lattice G (see figure 2) we get
P
n(FP1 =Km22 ) = m−1
j=1 [n(FP1 =Kj22 ) + n(FP1 =Kj21 ) + n(FP1 =Kj12 )] + n(FP1 =Km21 )+
n(FP1 =Km12 ) + n(FP1 =Km11 ).
(4)
According to (2), then (4) can be written as
P
n(FP1 =Km12 ) = m−1
j=1 [n(FP1 =Kj22 )+n(FP1 =Kj21 )+n(FP1 =Kj11 )]+2.n(FP1 =Km21 )+
m−2
2
.
By using (3) we have,
P
Pm−1
n(FP1 =Km22 ) = m−1
j=1 [n(FP1 =Kj22 )+2.n(FP1 =Kj21 )+n(FP1 =Kj11 )]+2.( j=1 [n(FP1 =Kj21 )+
n(FP1 =Kj11 )] + n(FP1 =Km11 )) + 2m−2 .
n(FP1 =Km22 ) =
Pm−1
j=1
[n(FP1 =Kj22 ) + 4.n(FP1 =Kj21 ) + 3.n(FP1 =Kj11 )] + 3.2m−2 .
The number of fuzzy subgroups of cuboid group
525
n(FP1 =Km22 ) = m−1
j=1 [n(FP1 =Kj22 )+4.n(FP1 =Kj21 )]+3.[n(FP1 =K111 )+n(FP1 =K211 )+
n(FP1 =K311 ) + ... + n(FP1 =K(m−1)11 )] + 3.2m−2 .
P
n(FP1 =Km22 ) =
Pm−1
j=1
[n(FP1 =Kj22 ) + 4.n(FP1 =Kj21 )] + 6.2m−2 .
n(FP1 =Km22 ) =
m−1
X
[n(FP1 =Kj22 ) + 4.n(FP1 =Kj21 )] + 3.2m−1 .
(5)
j=1
Now, consider,
Pm−1
j=1 n(FP1 =Kj21 ).
j=1 [n(FP1 =Kj21 ) = 4.
Pm−1
m−1
X
[n(FP1 =Kj21 ) = 4.[n(FP1 =K121 )+n(FP1 =K221 )+n(FP1 =K321 )+...+n(FP1 =K(m−1)21 )].
j=1
(6)
According Theorem 2.5 ii), for every i ∈ {1, 2, ...., (m − 1)}, we have
n(FP1 =Ki21 ) = 2i + (i − 3)2i−2 = (i + 1)2i−2 .
Hence, (6) can be written as,
Pm−1
0
1
2
m−3
).
j=1 [n(FP1 =Kj21 ) = 4(1 + 3.2 + 4.2 + 5.2 + ... + m.2
P
m
1
4
2
3
4
m
i
= 23 (2.2 + 3.2 + 4.2 + ... + m.2 ) = −1 + 2 i=1 i(2 ).
Since
Pm
Pm−1
4.n(FP1 =Kj21 ) = −1 + 12 (2 + (m − 1)2m+1 ) = (m − 1)2m .
j=1
i=1
i(2i ) = 2 + (m − 1)2m+1 (see [6];176), we conclude that,
From this, we can write (5) becomes
P
m
m−1
n(FP1 =Km22 ) = m−1
j=1 n(FP1 =Kj22 ) + (m − 1)2 + 3.2
Pm−1
= j=1 n(FP1 =Kj22 ) + (2m + 1)2m−1 .
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, 479-489.
[2] R. Sulaiman, Priyo Budi Prawoto, The number of fuzzy subgroups of
rectangle groups, International Journal of Algebra, 8 (2014), no. 1, 17-23.
http://dx.doi.org/10.12988/ija.2014.311121
[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, 53-58.
526
Raden Sulaiman
[4] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
http://dx.doi.org/10.1016/0022-247x(71)90199-5
[5] R. Sulaiman and Abd Ghafur Ahmad, Counting fuzzy subgroups of symmetric groups S2 , S3 and alternating group A4 , Journal of Quality Measurement and Analysis, 6 (2010), 57-63.
[6] R.P. Grimaldi, Discrete and Combinatorial Mathematics, Addison Wesley,
New York, 1999.
Received: October 5, 2015; Published: December 12, 2015