The Number of Fuzzy Subgroups of a finite Dihedral $D_{p^m q^n}$

Vol. 8, No. 1, 2015, 51-57

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 11 August 2015

www.researchmathsci.org

51

The Number of Fuzzy Subgroups of a finite

Dihedral

Amit Sehgal1, Sarita Sehgal2 and P.K. Sharma3 1

Govt. College, Birohar (Jhajjar), Haryana, India-124106 2Govt. College, Matanhail (Jhajjar), Haryana, India-124106

3

D.A.V. College, Jalandhar City, Punjab, India-144001

email: 1amit_sehgal_iit@yahoo.com; email: 3pksharma@davjalandhar.com

Received 31 July 2015; accepted 9 August 2015

Abstract. The main goal of this paper is to give formula for the number of fuzzy

subgroups of a finite dihedral group by using a recurrence relation

Keywords: Algebra; fuzzy subgroups; finite abelian p-groups; generating function;

Dihedral group

AMS Mathematics Subject Classification (2010): 20N25, 20E15

1. Introduction

One of the famous problems in fuzzy subgroup theory is to find the number of fuzzy subgroups of dihedral group. Some authors discussed the case of cyclic groups and finite Abelian group [1-3, 8-9]. In [10] Tă_rnă_{uceanu obtain formula for the fuzzy subgroups of} dihedral group and which was verified by Dabari, Saeedi and Farrokhi in [6]. In our earlier work [7], we have determined the number of subgroups of a finite abelian group of rank two. In this present paper, we establish a recurrence relation for the number of distinct fuzzy subgroups of dihedral group and solve this recurrence relation by using the concept which was already used by Tă_rnă_{uceanu and Bentea [2].}

2. Preliminaries

Let X be a fixed non-empty set. A Fuzzy Sets on X is a function from to [0,1]. A

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

(i) min , ,

(ii)

Clearly, max . For each 0,1!, the level subset corresponding to is defined as _" : 4! . A fuzzy subset is a fuzzy subgroup of G if and only if its level subsets are subgroups of G [5].

52

finite group and : - 0,1! is a fuzzy subgroup of G. Let , _., _/, . . , ₁ and assume that ' _.'. . . ' ₁. Then provides the following chain of subgroups of G ending with G

_"23 _"4 3 _"53 6 3 _" (1) Then for any and 7 1,2,3, … , *, we have _; ( _"<= _"<>2 with the assumption that _"? @. According to Volf [11], a necessary and sufficient condition for two fuzzy subgroups of G to be equivalent with respect to ~ is that they are same level fuzzy subgroups, that is, they determine the same chain of subgroups of type (1). Hence, there exists a bisection between the equivalence classes of fuzzy subgroups of G and the set of chains of subgroups of the group G, which end in G. If A_Bdenotes the number of all distinct fuzzy subgroups of G, then A_B is the number of chains of subgroups of length one of G ending in G plus the number of chains of subgroups of length more than one of the group G, which end in G.

AB 1 C ∑EFBAE

AB 1 C ∑H;IJ;1KJ ELIMB AEG *E, (2) where Iso(9) is the set of representative classes of subgroups of G and nH denotes the size

of the isomorphism class with representative H.

3. Dihedral group

The group ₁ is known as Dihedral group of order 2n, this group is generated by two elements: a rotation y of order n and a reflection x of order 2. Under these notations, we have, ₁N , | . 1, 1 1, '.

Theorem 3.1. [12] Every subgroup of ₁ is cyclic or dihedral. A complete list of subgroups is as follows:-

(i) Cyclic subgroups N H ' , where d|n

(ii) Dihedral subgroups N H, ;', where d|n and 0 P 7 P + = 1

Here cyclic subgroup N H' ~Q_1/H and N H, ; ' ~_1/H . As follows form Theorem 3.1, we have following results:

Theorem 3.2. Number of subgroups of group S_TU are VW ~XTY Z[\]\ Y ^, U

_____ TUY _~S

TY Z[\]\ Y ^, U_____`

Theorem 3.3. Number of subgroups of group S_Ta_bU are

cTW ~Xa_bU TYbd Z[\]\ Y ^, a______ eUf d ^,U_____

TY_bd ~STY_bd Z[\]\ Y ^, a______ eUf d ^,U_____

`

Now we contract recurrence relations for number of fuzzy subgroups of group Q,

)*+ .

Theorem 3.4. The number of all distinct fuzzy subgroups of S_Ta_bU is

ga

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Tka_lmkaWno_aW mr =WlnpnW_gpno_bp_senp

e,p t sb C g =WbWTt e

= ∑ jka_lmkano_a mblno_TanW_{= gk}anW

l mkanoa m C

∞

engpnlnoqUW

Tka_lmkaWno_aW mr =WlnpnW_gpno_bpnW_senp

e,p t sb C g =WbWTt e

uv

4. Construction of recurrence relations

Theorem 4.1. If wkX_Ta_bUm denotes number of fuzzy subgroups of group X_Ta_bU, then we can establish relation as

wkXTabUm = gwkXTa_bU>Wm = gwkX_Ta>W_bUm C gwX_Ta>W_bU>W ^.

Proof: We know that, the number of subgroups of group X_Ta_bU which are isomorphic to group X_Tl_b[ is 1 for 0 P x P * )*+ 0 P y P z

By using (2), we have AQ 1 C ∑_€q~ ∑_}q~1AQ{|C ∑_€q~ AQ{ (3)

Change n into n-1 in (3), we get

AQ>2 1 C ∑_€q~ ∑_}q~1.AQ{|C ∑_€q~ AQ{>2 (4) From (3) and (4), we get

AQ = 2AQ>2 ∑_€q~ AQ{>2 (5) Change m into m-1 in (5), we get

AQ>2 = 2AQ>2>2 ∑._€q~ AQ{>2 (6) From (5) and (6), we get

AkQm = 2AkQ>2m = 2AkQ>2m C 2AQ>2>2 0 (7)

Theorem 4.2. If wkS_Tam denotes number of fuzzy subgroups of groupS_Ta, then we can establish relation as

wkSTam = gTwkS_Ta>Wm gwkX_Tam = gTwkX_Ta>Wm Proof: By using (2) and Theorem 3.2, we have

Akm 1 C ∑ AsQ1_;q~ <tC ∑1_;q~ 1;As<t (8)

AkQm 1 C ∑1_;q~AsQ<t (9) (8)-(9), we get

Akm = 2AkQm ∑1_;q~ 1;As<t (10) Change n to n-1 in (10), we get

Ak>2m = 2AkQ>2m ∑1._;q~1;As<t (11) (10)-p (11), we get

Akm = 2Ak>2m 2AkQm = 2AkQ>2m (12)

Theorem 4.3. If wkS_Ta_bUm denotes number of fuzzy subgroups of groupS_Ta_bU, then we can establish relation as

wkSTa_bUm = gbwkS_Ta_bU>Wm = gTwkS_Ta>W_bUm C gTbwkS_Ta>W_bU>Wm

54 Proof: By using (2) and Theorem 3.2, we have

Akm

1 C ∑ ∑_;q~ 1_ƒq~AsQ<‚tC ∑_;q~∑1_ƒq~;„1ƒAs<‚tC ∑1_ƒq~„1ƒAs‚t (13)

AkQm 1 C ∑_;q~∑_ƒq~1 AsQ<‚tC ∑1_ƒq~AsQ‚t (14)

(13)-(14), we get

Akm = 2AkQm ∑_;q~ ∑1_ƒq~;„1ƒAs<‚tC ∑1_ƒq~„1ƒAs‚t

(15) Change n to n-1 in (15), we get

Ak>2m = 2AkQ>2m

∑ ∑ ;_„1ƒ_As

<‚t

1 ƒq~ 

;q~ C ∑1._ƒq~„1ƒAs‚t (16) (15)-q (16), we get

Akm = 2AkQm = 2„Ak>2m C 2„AkQ>2m ∑_;q~ ;As<t

(17) Change m to m-1 in (17), we get

Ak>2m = 2AkQ>2m = 2„Ak>2>2m C 2„AkQ>2>2m

∑ ;_As

<t

.

;q~ (18)

(17)-p(18), we get

Akm = 2AkQm = 2„Ak>2m C 2„AkQ>2m = 2Ak>2m C

2AkQ>2m C 2„Ak>2>2m = 2„AkQ>2>2m 0

Akm = 2„Ak>2m = 2Ak>2m C 2„Ak>2>2m 2AkQm =

2„AkQ>2m = 2AkQ>2m C 2„AkQ>2>2m (19) Now we find solution of these recurrence relations which satisfies required initial

condition.

5. Solution of recurrence relation

Multiply both sides of (19) by 1 and summation n from 1 to ∞, we get

∑∞_1qAkm 1= 2„ ∑∞_1qAk>2m 1= 2 ∑∞_1qAk>2m 1C

2„ ∑∞_1qAk>2>2m 1 2 ∑∞ AkQm

1q 1= 2„ ∑∞_1qAkQ>2m 1=

2 ∑∞_1qAkQ>2m 1C 2„ ∑∞_1qAkQ>2>2m 1 Take ∑∞ Akm

1q~ 1 … and ∑∞ AkQm

1q~ 1 †, we get

…= Akm = 2„ …_= 2…_= Ak>2m C 2„ …_ 2†_=

AkQm = 2„ †_= 2†_= AkQ>2m C 2„ †_

1 = 2„ …= 21 = „ …= sAkm = 2Ak>2mt 2 = 2„ †_=

21 = „ †= 2AkQm = AkQ>2m!.

55

1 = 2„ …= 21 = „ …= 22 =  2 = 2„ †= 21 = „ †= 22= 2!1 = 2„ …= 21 = „ …

2 = 2„ †= 21 = „ † (20)

Now we find value †_, multiply both sides of (8) by 1 and summation n from 1 to ∞, we get

†= AkQm = 2 †_= 2†_= AkQ>2m C 2 †_ 0

†1 = 2 = †2 = 2 AkQm = 2AkQ>2m 2= 2 0

†=..‡_.‡† 0.

The general solution of above equation is †_ ˆ..‡_.‡

We know that AkQ?m 21

Then †_~ ∑∞ 21

1q~ 1_.‡ Hence, †_{ .‡} ..‡_.‡ Put the value of †_, we get

2 = 2„ †= 21 = „ † s..‡_.‡ts‡..‡n.‡_.‡‡ t Hence, (20) becomes

1 = 2„ …= 21 = „ … s..‡_.‡t 

s‡..‡n.‡_.‡‡ t

Then …_=.‡_{.‡ …} s..‡_.‡ts‡..‡n.‡_.‡‡.‡t (21)

C.F of the recurrence relation ‰ is ˆ.‡_.‡

P.I of the recurrence relation (21) is … s..‡_.‡t Now we find B

… s..‡_.‡t=.‡_.‡ … s..‡_.‡t s..‡_.‡ts‡..‡n.‡_.‡‡.‡t

…1 =‡.‡_.‡‡! s‡..‡n.‡_.‡‡.‡t

… s_{.‡nn..‡n..‡}‡..‡n.‡ 4t

Then, the general solution of (21) is

… ˆ.‡_.‡C s_{.‡n.n.‡n..‡}‡..‡n.‡ 4t s..‡_.‡t



Put m=0 in ∑∞ Akm

1q~ 1 …~

…~ ∑ .

Š2_n.

∞

1q~ 1 i_.‡ C_.‡.u i_.‡.‡.‡ u Put m=0 in (10), we get

…~ ˆ C s_{.‡n.nn.‡n..‡}‡..‡n.‡ 4t

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ˆ i_{.‡n.n.‡n..‡}‡ 4u

Hence

…

i_{.‡n.n.‡n..‡}‡ ₄u .‡_.‡_C

s_{.‡n.n.‡n..‡}‡..‡n.‡ ₄t s..‡_.‡t

… .

_‡

n.n.‡n..‡4‹=n ‡

.‡Š2C 2 s_.‡‡t

n

=  s_.‡‡tŒ

…

._‡

i∑∞nŽn€nJq~jk€mknJ m„€nJn= 2kn€ mknJ m C k_€mknJ_ mr =1€nŽn₂ŽnJ_„Ž_snŽ

,Ž t s„ C 2 =t

 _€nJnn.Ž u But …_ ∑∞ Akm

1q~ 1 , Hence Akm ‘‘77*’ ‘ 1 7* …_

Akm

.

hi∑∞n.Žn€nJq1jk€mknJ m„€nJn= 2kn€ mknJ m C k_€mknJ_ mr =1€nŽn₂ŽnJ_„Ž_snŽ

,Ž t s„ C 2 =t 

= ∑ jk_€mknJ_ m„€nJ_n_{= 2k}n

€ mknJ m C

∞

n.Žn€nJq1 k

€mknJ mr =1€nŽn2ŽnJ„ŽnsnŽ,Ž t s„ C 2 =t 

uv (22)

Corollary [6, 10]. The number of all distinct fuzzy subgroups of S_Ta_bW is

ga

TWgja C gTangb = a C “TanWb C gTang= TanWC a C gTg=

“a C ”T C ga C •!

Proof. Put n=1 in equation (22), we get desired result.

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