Vol 3, No 4 (2012)

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Available online through

ISSN 2229 – 5046

International Journal of Mathematical Archive- 3 (4), April – 2012 1712

THE DETERMINATION OF THE NUMBER OF DISTINCTFUZZY SUBGROUPS

OF GROUP

1 2 ... n

p p p

Z

_{× × ×}

AND THEDIHEDRAL GROUP

1 2 2 p p ... p_n

D

_{× × × ×}

HASSAN NARAGHI

1*

& HOSEIN NARAGHI

2

1

Department of Mathematics, Islamic Azad University, Ashtian Branch, P.O. Box 39618-13347, Ashtian, Iran

2

Department of Mathematics, Payame Noor University, Ray Branch, Iran

E-mail: 1[email protected],

(Received on: 03-04-12; Accepted on: 27-04-12)

________________________________________________________________________________________________

ABSTRACT

I

n this paper, we use the natural equivalence of fuzzy subgroups studied by Iranmanesh and Naraghi [3] to determine the number of distinct fuzzy subgroups of some finite groups. We focus on the determination of the number of distinct fuzzy subgroups of group

1 2 ... n

p p p

Z × × × and the dihedral group D2× × × ×p1 p2 ... pnusing this equivalence relation.

2000 Mathematics Subject Classification: 20N25.

Key words and phrases: Dihedral group, Equivalence relation, Fuzzy subgroups.

________________________________________________________________________________________________

1. INTRODUCTION

Zadeh introduced the notion of fuzzy sets and fuzzy set operations, in his classic paper [15] of 1965. In an analogous application with groups, Rosenfeld [13] formulated the elements of a theory of fuzzy groups. One of the most important problems of fuzzy group theory is to classify the fuzzy subgroups of a finite group. This topic has enjoyed a rapid evolutionin the last years. Many papers have treated the particular case of finite cyclic groups. Thus, in [8] the number of distinct fuzzy subgroups of a finite cyclic group of square-free order is determined, while [11, 12, 14] deal with this number for cyclic groups of order pnqm (p, q primes). In the present paper we establish the recurrence relation verified by the number of distinct fuzzy subgroups of group

1 2 ... n

p p p

Z _{× × ×} and the dihedral group

1 2

2 p p ... pn

D× × × × such that p1,p2,…,pn

are distinct primes.

2. PRELIMINARIES

First of all, we present some basic notions and results of fuzzy sub group theory (for more details, see [4, 7, and 3]).

The dihedral group of order 2n, for n

≥

2, denoted byD₂_n. A fuzzy sub set of a set X is a mapping

[ ]

:X 0,1 .

µ → Fuzzy subset

µ

of a group G is called a fuzzy subgroup of G if:

1

(G ) µ(xy) ≥ (x) µ ∧ (y) for all x; y µ ∈ G;

-1 2

(G ) (x ) µ ≥µ(x) for all x ∈ G

The set of all fuzzy subgroup of a group G is denoted by F (G).

Definition 2.1: Let G be a group andµ∈F G

( )

. The set of

{

x∈G µ

( )

x >0

}

is called the support of

µ

and denoted by supp

µ

.

Let G be a group andµ∈F G

( )

. We shall write Imµfor the image set of

µ

and F

µ

for the family

{

µt |t∈Imµ

}

. ________________________________________________________________________________________________

*Corresponding author: 1

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Theorem 2.2: Let G be a fuzzy group. If

µ

is a fuzzy subset of G, then µ∈F G

( )

if and only if for all µ_t ∈F_µ, µ_t is a subgroup of G.

Let F G₁( ) be the set of all fuzzy subgroups µ of G such thatµ

( )

e =1, and let ~ R be an equivalence relation

onF G₁( ). We denote the set

{

v∈F G₁

( )

|v ~_R µ

}

by ~R

µ

and the set | 1

( )

~R

F G

µ µ

 

∈

 

 by

( )

1

~R

F G

.

Definition 2.3: Let G be a group, andµ,v∈F G1

( )

.

µ

is equivalent to

v

, written as

µ

~

v

if

(1)µ

( )

x >µ

( )

y ⇔v x

( )

>v y

( )

for all x y, ∈G. (2) µ

( )

x = ⇔0 v x

( )

=0for allx∈G.

The number of the equivalence classes ~ on F G₁( )is denoted bys G

( )

. We mean the number of distinct fuzzy subgroups of G is s (G).

Theorem 2.4: Let G be a finite group. The number of distinct fuzzy subgroups of G such that their support is exactly

equal to G is

( )

1 2

s G +

.

Let G be a finite group. The number of distinct fuzzy subgroups of G such that their support is exactly equal to G is denoted bys∗

( )

G .

Theorem 2.5: [3] Let G be a finite group. Then the number of distinct fuzzy subgroups of G such that their support is

exactly a subgroup of G is

( )

1. 2

s G −

Theorem 2.6: [3] Let G be a finite group and H be a subgroup of G. Then the number of distinct fuzzy subgroups of G

such that their support is exactly equal to H is

( )

1. 2

s H +

Corollary 2.7: [3] Let G be a finite group and H be a subgroup of G. Then the number of distinct fuzzy subgroups of G

such that their support is exactly a subgroup of H is

( )

1 2

s H −

.

Proposition 2.8: [6] Let

n

∈

N

. Then there are 2n+1−1distinct equivalence classes of fuzzy subgroups ofZ_pn.

3. THE NUMBER OF THE DISTINCT FUZZY SUBGROUPS OF THE ABELIANGROUP

1 2 ... n

p p p

Z _{× × ×}

In this section, we characterize fuzzy subgroups of the abelian group

1 2 ... n

p p p

Z × × such that p p1, 2,...,pnare distinct

primes numbers (n > 1).

Proposition 3.1: Suppose that p and q are distinct primes. Then there are11 distinct equivalence classes of fuzzy subgroups of Zpq.

Proof: See Theorem 8.2.4 of [6].

Proposition 3.2: Suppose that p, q and r are distinct primes. Then there are 51 distinct equivalence classes of fuzzy subgroups of Zpqr.

Proof: We know that Zpqr has the following maximal chains:

{ }

0 pqr pq p

Z ⊃Z ⊃Z ⊃ ,Z_pqr ⊃Z_pq ⊃Z_q ⊃

{ }

0 ,Z_pqr ⊃Z_pr ⊃Z_p ⊃

{ }

0 ,

{ }

0 pqr pr r

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All of subgroups of the groupZpqr are Zpq,Zpr,Zqr,Zp,Zq,Zrand

{ }

0 .Thus

( )

1

_{( )}

_{{ }}

_{( ) ( ) ( ) ( ) ( )}

_{( )}

0

2 pq pr qr p q r

s G

s∗ s∗ Z s∗ Z s∗ Z s∗ Z s∗ Z s∗ Z

−

= + + + + + + ,

therefore

( )

1

_{( )}

_{( ) ( )}

1 3 3 1 3 6 3 2 2 5

2 pq p

s G

s∗ Z s∗ Z

−

= + + = + + = .

Hence

( )

51

s G = .

Theorem 3.3: Suppose that p p₁, ₂,...,p_nare distinct primes. If

1 2 ... n

p p p

G =Z _{× × ×} and n>1, then

( )

1

( i ) 2 1

j j n n p i n

s G s Z

i = − =   = _{ } + + ∏  

∑

.

Proof: Denote

∏

_k=

{

pi1× ×... pik | ,...,i1 ik ∈

{

1,...,n

}

,i1< <... ik

}

, k =1, 2,...,n. We know that G =Zp1× × ×p2 ...pn has

the following maximal chains each of which can be identified with the chain

{ }

1 .... 1 0

n n

Zπ ⊃Zπ− ⊃ ⊃Zπ ⊃ such

thatπi ∈ Πi , for alli∈

{

1,...,n

}

. For all

(

)

! 1,.., ,

! !

n n

i n

i i n i

 

= _{ }=

−

  is the number of subgroups of the group G

as

i

Zπ . Therefore by theorem 2.5,

( )

{ }

( )

1 1 1 0 2 i n i

s G n

s s Z

i π − ∗ ∗ = −   = + _{ }  

∑

and hence

( )

1 1 2 3 i n i n

s G s Z

i π − ∗ =   = _{ } +  

∑

.By

theorem 2.4,

( )

1 1 2 1 i n n i n s G s Z

i π − =   = _{ } + +  

∑

.

4. THE NUMBER OF DISTINCT FUZZY SUBGROUPS OF THE DIHEDRALGROUP

1 2 2 p p ... pn

D_{× × × ×}

In this section, we determine the number of distinct fuzzy subgroups ofthe dihedral group

1 2

2 p p ... pn

D× × × × such that

1, 2,..., n

p p p are odd distinct primes.

Theorem 4.1: Suppose that p is a prime and

p

≥

3

. If G is the dihedral group of order 2p, then s(G) = 4p + 7.

Proof: We know that D2p has the following maximal chains:

{ }

2p p 0

D ⊃Z ⊃ and D2p ⊃Z2 ⊃

{ }

0 whose the number is p. Now 2 is the number of distinct fuzzy subgroups whose

support is Zp, 21pis the number of distinct fuzzy subgroups whose support isZ2, and 0

2 is the number of fuzzy

subgroups whose support is

{ }

0 . Thus

( )

1 2 2 1 2

s G

p

−

= + + , therefores G

( )

=4p+7.

Theorem 4.2: Suppose that p and q are odd distinct primes. If G is the dihedral group of order 2pq, then

( )

12 8

(

)

23

s G = pq+ p+q + .

Proof: We know that D₂_pq has the following maximal chains:

{ }

2pq 2p 2 0

D ⊃D ⊃Z ⊃ ,D2pq ⊃D2p ⊃Zp ⊃

{ }

0 ,D2pq ⊃D2q ⊃Z2 ⊃

{ }

0 ,D2pq ⊃D2q ⊃Zq ⊃

{ }

0 ,

{ }

2pq pq p 0

D ⊃D ⊃Z ⊃ ,D2pq ⊃Dpq ⊃Zq ⊃

{ }

0 . Clearly, pq is the number of the subgroups Z2of the dihedral

groupD2pq, q is the number of the subgroups D2pand p is the number of the subgroups D2qof the dihedral group D2pq

and the dihedral group D2pq has just one subgroup asZqp,Zq,Zp. So that

( )

_{( )}

_{{ }}

_{( )}

_{( ) ( ) ( ) ( )}

2 2 2

1 0

2 p q pq p q

s G

s∗ pqs∗ Z qs∗ D ps∗ D s∗ Z s∗ Z s∗ Z

−

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Thus

( )

1

₍

₎

₍

₎

1 2 2 4 2 4 6 2 2

2

s G

pq q p p q

−

= + + + + + + + + ,

therefore

( )

12 8

(

)

23

s G = pq+ p+q + .

Table 1: The number of distinct fuzzy subgroups of dihedral group D₂_pqfor some selected primes.

2pq

G =D s(G) 3, 5

p= q = 267 3, 7

p = q = 355 3, 11

p = q = 531 3, 13

p= q= 619 13, 17

p= q= 2915

Theorem 4.3: Suppose that p, q and r are odd distinct primes. If G is the dihedral group of order 2pqr, then

( )

52 24

(

)

24

(

)

103

s G = pqr+ pq+pr+qr + p+ +q r + .

Proof: We have

2 1

2 , | 1,

n n

D =<x y x =y = yxy =x− >.

It is well Known that for every divisor r of n, D2npossesses a subgroup isomorphic toZr, namely 0

n

r r

H =<x >and

n

r subgroups isomorphic toD2r, namely

1

, , 1, 2,...,

n

r r i

i

n H x x y i

r −

=< > = . We know that D₂_pqrhas the following

maximal chains each of which can be identified with the Chain,

{ }

2pqr 2pq 2p p 0

D ⊃D ⊃D ⊃Z ⊃ ,D2pqr ⊃D2pq ⊃D2q ⊃Zq ⊃

{ }

0 ,D2pqr ⊃D2pr ⊃D2p ⊃Zp ⊃

{ }

0 ,

{ }

2pqr 2pr 2r r 0

D ⊃D ⊃D ⊃Z ⊃ ,D2pqr ⊃D2qr ⊃D2q ⊃Zq ⊃

{ }

0 ,D2pqr ⊃D2qr ⊃D2r ⊃Zr ⊃

{ }

0 ,

{ }

2pqr 2pq 2p 2 0

D ⊃D ⊃D ⊃Z ⊃ ,D2pqr ⊃D2pq ⊃D2q ⊃Z2 ⊃

{ }

0 ,D2pqr ⊃D2pr ⊃D2p ⊃Z2 ⊃

{ }

0 ,

{ }

2pqr 2pr 2r 2 0

D ⊃D ⊃D ⊃Z ⊃ ,D2pqr ⊃D2qr ⊃D2q ⊃Z2⊃

{ }

0 ,D2pqr ⊃D2qr ⊃D2r ⊃Z2 ⊃

{ }

0 ,

{ }

2pqr pqr pq p 0

D ⊃Z ⊃Z ⊃Z ⊃ ,D₂_pqr ⊃Z_pqr ⊃Z_pq ⊃Z_q ⊃

{ }

0 ,D₂_pqr ⊃Z_pqr ⊃Z_pr ⊃Z_p ⊃

{ }

0 ,

{ }

2pqr pqr pr r 0

D ⊃Z ⊃Z ⊃Z ⊃ ,D₂_pqr ⊃Z_pqr ⊃Z_rq ⊃Z_q ⊃

{ }

0 andD₂_pqr ⊃Z_pqr ⊃Z_rq ⊃Z_r ⊃

{ }

0 .

Clearly, pqr is the number of subgroups Z₂of the dihedral groupD₂_pqr, qr is the number of the subgroupsD₂_p, pr is the number of the subgroupsD₂_q, pq is the number of the subgroups D₂_rand p is the number of the subgroupsD₂_qr, q is the number of the subgroupsD₂_prand r is the number of the subgroups D₂_pqof the dihedral groupD₂_pqr and the dihedral group D2pqr has just one subgroup asZpqr, Zpq,Zpr, Zqr, Zq, Zp, Zr.So that

( )

_{( )}

_{{ }}

_{( )}

_{( ) ( )}

_{( )}

_{( ) ( ) ( ) ( )}

2

1 0

2 p q r pq pr qr pqr

s G

s∗ p qs∗ Zr s∗ Z s∗ Z s∗ Z s∗ Z s∗ Z s∗ Z s∗ Z

−

= + + + + + + + +

qrs

( )

D2p prs

( )

D2q pqs

( )

D2r ps

( )

D2qr qs

(

D2pr

)

rs

(

D2pq

)

∗ ∗ ∗ ∗ ∗ ∗

+ + + + + + ,

Therefore

( )

1

_{( )}

₍

₎

₍

₎

₍

₎

1 2 3 2 3 6 26 2 4 2 4 2 4

2 p

s G

pqr Z qr p pr q pq r

−

= + + + + + + + + + +

(

)

(

12 8 24

)

(

12 8

(

)

24

)

(

12 8

(

)

24

)

2 2 2

qr q r pr p r pq p q

p + + + q + + + r + + +

+ + + .

Thus

( )

52 24

(

)

24

(

)

103

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Table 2: The number of distinct fuzzy subgroups the dihedral group

D

₂_pqrfor some selected primes.

2pqr

G =D s(G)

3, 5, 7

p= q= r= 7627

3, 7, 11

p= q= r= 15763

5, 7, 13

p= q = r= 28947

7, 13, 17

p= q= r= 91779

13, 17, 19

p= q= r = 238611

Theorem 4.4: Suppose that p p₁, ₂,...,p_nare odd distinct primes and P = ×2 p₁×p₂× ×... p_n. IfG =D_pand n>1, then

( )

1

1 2

1 | ,2

2 ( ) (2 3) 2 2

2 2 i j j n n n t p

i t P t P

n P P

s G P s Z s D

i =

+

= < <

 

= + _{ } + + − + +

∏  

∑

.

Proof: We have

2 1

2 , | 1, .

n n

D =<x y x =y = yxy =x− >

It is well Known that for every divisor r of n, D₂_npossesses a sub group isomorphic toZ_r, namely 2 0

n r

H

=<

x

>

and

n

r subgroups isomorphic toD2r, namely

1

, , 1, 2,..., .

n

r r i

i

n H x x y i

r −

=< > = Let

{

}

{

1 .... k | ,....,1 1,...., , 1 ...

}

k pi pi i ik n i ik

Π = × × ∈ < < ,k =1, 2,...,n.

We know that G=D_P has the following maximal chains each be identified with the chain

{ }

1 1 1

2 n 2 n

...

2

0 D

_π

⊃

D

_π ₋

⊃

D

_π

⊃

Z

_π

⊃

,

{ }

1 1

2 n n n .... 0

Dπ ⊃Zπ ⊃Zπ− ⊃ ⊃Zπ ⊃ ,

{ }

1 1

2 n 2 n ... 2 2 0

D π ⊃Dπ₋ ⊃ ⊃Dπ ⊃Z ⊃ ,

such thatπi ∈ Πi for all i ∈

{

1,...,n

}

. Now

2

P

is the number of subgroups of the group G asZ2,and for

alli =1,...,n,

(

!

)

! !

n n

i i n i

 ₌

  ₋

  is the number of subgroups of the group G asZπi . Also

2

P

t is the number of subgroups

of the group G asD₂_t,for every divisor t of 2

P

. Therefore by theorem 2.5,

( )

_{( )}

_{{ }}

_{( )}

₍

₎

2 2

1 | ,2

1 0

2 2 i 2

n

t

i t P t P

s G P n P

s s Z s Z s D

i π t

∗ ∗ ∗ ∗

= < <

−  

= + + _{ } +

 

∑

,

Then

( )

(

2

)

(

2

)

1 | ,2

2 2 ( ) 3

i

n

t t

i t P t P

n P

s G P s Z P s D s D

i π t

∗ ∗

= < <

 

= + _{ } + + +

 

∑

,

Thus

( )

(

2

)

1 | ,2

1

2 ( ) 2 2

2

i

n

t n

i t P t P

s D

n P

s G P s Z

i π t

= < <

+   = + _{ } + + +  

∑

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