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A note on an equivalence relation on fuzzy subgroups
ZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 95 (1998) 243-247
Short Communication
A note on an equivalence relation on fuzzy subgroups 1 Yunjie Zhang*, Kaiqi Zou Department of Basic Science, Dalian Maritime University, Dalian, 116026, People's Republic of China Received May 1996; revised April 1997
Abstract The purpose of this paper is to discuss the following three problems: (1) the number of equivalent classes of fuzzy subgroups of a group G, (2) the relationship between the number of equivalent classes of fuzzy subgroups and the order of a cyclic group G, (3) the relationship between the order of fuzzy subgroups and the order of a cyclic group G, (4) a classification of a group, using membership values of fuzzy subgroups, is given. © 1998 Elsevier Science B.V.
Keywords: Group; Fuzzy subgroup; Equivalent class; Classification
1. Preliminaries
The set of all fuzzy subgroups of a group G will be denoted by F(G).
We recall some definitions and results that will be used later. I f S is a set, then a mapping # : S --* [0, 1] is called a fuzzy subset of S. Let # be a fuzzy subset of S, by an 2-cut #x o f # we mean the following crisp subset of S:
#~= {xlx~S, #(x)>l,~, ,~[o,1]}. Clearly, 2t ~<).2 implies #~, _D#~2. Definition 1.1. Let G be a group. Then a fuzzy subset # of G is said to be a fuzzy subgroup of G if
Definition 1.2. Let G be a group. A # C F(G) is called a normal fuzzy subgroup of G if
g(xyx-1)>...#(y),
Vx, yEG.
Definition 1.3. Let G be a group. A # EF(G) is called a characteristic fuzzy subgroup of G if Vtp E Aut G, g(tp(x)) >~#(x),
Vx E G,
where Aut G is the automorphism group of G.
Theorem 1.1. Let G be a group and #EF(G). Then
#(xy)>~min{~(x),#(y)}, # ( x - l ) / > #(x),
Vx, yEG,
VxEG.
* Corresponding author. I Supported by science foundation of Dalian Maritime University. 0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 1 8 5 - 1
l~ is a normal fuzzy subgroup of G if and only if #(xyx -1) = #(y),
Vx, yEG.
Let G be a group, #EF(G). We shah write Im # for the image set of # and Fu for the family {#~ 12EIm#}.
244
Y. Zhang, K. Zou/Fuzzy Sets and Systems 95 (1998) 243~47
Theorem 1.2. Let G be a group. (1) I f # is a fuzzy subset of G, then I~CF(G) if and
only if V#~ E Fu, #~ ¢ ~, #~ is a subgroup of G. (2) V # E F ( G ) , p is a normal (resp. characteristic) fuzzy subgroup if andonly if Vl~ E F~, I~ ¢ qJ, #~. is a normal (resp. characteristic) subgroup of G (see [3, 5]). (3) I f HoCH1CH2C ... C H r = G is a chain of subgroups, then 3~CF(G) such that F~= {Ho,H1,H2 . . . . . Hr} (see [6, Theorem 10.2.7]). Definition 1.4 (Dixit [1, Definition 3.2]). Let /,,vE F(G). We say that/t is equivalent v, written as/t ,-~ v, ifF u =&.
then I&l =
where
s, = {a},
& = {{G,{e}I,{G,H~} ..... { G , H , _ , }},
&+, = {{e},H1 . . . . . H , _ , , G}. Consequently, rc = ~ = 0 C~ = 2". Case 2: V/-/,.,Hj (i,j ¢O,n), Hi ¢ I-Ij and & g Hi. Let
Ri = {{e},Hi, G},
i= 1,2 . . . . . n - 1
and
Clearly, " ~ " is an equivalent relation on F ( G ) , and F(G) can be classed under the relation " ~ " . The number of equivalent classes will be denoted by re.
Lemma 1.1. L e t G b e a finite group. Then G is a cyclic group if and only if V d l o ( G ), there exists precisely a subgroup of order d (see [7, Theorem 1.4.5 and Corollary]). Definition 1.5 (Ldszl6 [4, Definition 1]). The order of the set of membership values of a fuzzy set/z is called the order o f / t and is denoted by II,I
Theorem 1.3 (Lfiszl6 [4, Corollary 2]). I f in a group G the length of the longest maximal subgroup chain with endpoints {e} and G is n, the order of any fuzzy subgroup of G cannot be greater than n+l.
&={F~C_RilltEF(G) and l14 =k},
k=1,2,3.
Since {G} = S,1 and {{e}, G} cS/2 for Vi (i = 1,2 . . . . . n - 1), from Case I, it follows that rc = ( 2 2 - 2) (n-1)+2=2n. Case 3: We can assume, without loss of generality, that
I{e}
c, = {H,,H2,...,H,,
CH, CH2C.. C2 =
CH,,CG},
{Hn,+,,Hn,+2,...,Hn2 I { e } C H,,+I C . . ' C H,2 C G },
Cm = {mn,n ,+l,Hnm_,+2 . . . . .
O.l{e}
CH,~_,+~ C...CHoC G}, 2. On equivalent classes of fuzzy subgroup Theorem 2.1. Let the number of proper subgroups of a group G be n. Then 2n<~ra <~2". Proof. Let {{e}=Ho,HbH2 ..... Hn=G}
be the set of all subgroups of G. This can happen in three cases: Casel: { e } = H o C H 1 C H 2 C ... CHn = G. From Theorem 1.2(3) and Definition 1.4, we have G C_FI, and l~<]#l~
&--{~ll#l--k},
k=l,2 ..... n+l,
where nl < n 2 < " ' "
Hj gSHk
and
Hk ~ Hj
(k=nm-i + 1,nm-I + 2 . . . . . ni). Similar to Case 1, we have rG = 2 n' + (2 n2
n,+l
_
2) + .-. + (2n-n'-'+l -- 2).
Clearly, 2n ~<2 nl q- ( 2 n 2 - n , + 1 _ 2 ) + - - - + (2 n - n ' - , +~ _ 2) ~<2 n. This completes the proof. []
Y. Zhano, K ZoulFuzzy Sets and Systems 95 (1998) 243-247
Corollary 2.1. I f a group G & a finite, then r6 is finite.
Proof. By Theorem 2.1 and [2, Theorem 1.5.4].
[]
3. ra and cyclic group Theorem 3.1. Let G = ( a ) be a cyclic group and p a prime. (1) I f o ( G ) = p n, then r 6 = 2 n. (2) I f rG = 2 n, then we have either o ( G ) = pn or o ( G ) = pSq, where 1 <~s<~n - 1, both p and q are distinct prime. Proof. (1) Since G = ( a ) be a finite cyclic group, o(G) is equal to order of a. By Lemma 1.1 and Sylow's first theorem (see [2, Theorem 4.2.1]), there must exist a chain {e} :
theorem, there is at least a chain {e} :
Corollary 2.2. A group G is a cyclic group of prime order if and only if re = 2.
(a p") C (a p"-') C " " C (a p2) C (a p) C G
of subgroups of G, a n d {(aP"),(a p" ') . . . . . (ap2), (aP)} is precisely the set of all proper subgroups of G. Consequently, by Theorem 2.1, rG = 2 n. (2) By Corollary 2.1, G = ( a ) is a finite cyclic group, and o(G) is equal to order of a. Let the order of a be m, then we have two cases. Case 1: n = 1. By Corollary 2.2, m is a prime and o(G) = p, where p = m. Case 2: n > 1. By Corollary 2.2, m is not a prime and m = pSq, where p is a prime and PXq, both s and q are positive integers. Case 2.1: q = 1. By Lemma 1.1 and Sylow's first theorem, there exists a chain
245
(a pnq) C (a pn) C (a pn-' ) C " " C (a pz ) C (a p) C G
of subgroups of G. Consequently, rG~>2 n+l by Theorem 2.1 and this contradicts that rc = 2 n. This completes the proof. []
Theorem 3.2. Let G be a group and o( G) = pn. Then G is a cyclic group if and only if re = 2 n. Proof. The necessity is straightforward by Theorem 3.1(1 ). Now, we prove the sufficiency. By Sylow's first theorem, there is a chain {e} =H1 c H 2 c ' " C H n C G of subgroups of G and the order of Hi is pn-i+l ( i = 1 , 2 . . . . . n). Suppose that there exists a subgroup H / ¢ H i and the order of HI is also p,-i+l, then by Theorem 2.1, rG > 2" and this contradicts that r c = 2 n. Therefore, ViC{1,2 . . . . . n}, G has a unique subgroup Hi of order pi. I t follows that G is a cyclic group by Lemma 1.1. With this our proof is complete. [] Let NF(G) (resp. CF(G)) denote the set of all normal (resp. characteristic) fuzzy subgroups of a group G. Then, by Definition 1.4, NF(G) (resp. CF(G)) can be classed under the relation " ~ " . The number of equivalent classes in NF(G) (resp. CF(G)) will be denoted by ru (resp. rc). It is easy to prove that
Theorem 3.3. ( 1) A group G is a simple group if and only if r N = 2. (2) A group G is a characteristically simple group if and only if rc = 2.
{e} =(aP~) c (aP~-' ) C . . . c (aPZ) c (aP) C G
4. I#l and cyclic group of subgroups of G and { ( a P ' ) l i = l , 2 . . . . . s} is precisely the set of all proper subgroups of G. By Theorem 2.1, if s>~n+ 1, then rG~2 n+~ and this contradicts that r6 = 2 n; if s ~
Theorem 4.1. Let G be a cyclic group. Then G & a finite cyclic group if and only if Vp E F(G), [p[ is finite. Proof. The necessity is straightforward by Theorem 1.3. Now, we prove the sufficiency.
Y. Zhano, K. ZoulFuzzy Sets and Systems 95 (1998) 243-247
246
Suppose that G is a infinite cyclic group. Let G --- (a). Then the order o f a is infinite• Consequently, there is a chain
o f subgroups o f G. Let/2 : G ~ [0, 1] such that Vx E G,
{e} C " "
/2(x ) =
C (a 2" ) C ' "
C (a 4) C (a 2 ) C G
o f subgroups o f G. L e t / 2 " G ~ [0, 1] such that Vx E G,
1,
x E {e},
es,
XE (a p~-') -- (a p~),
Cl,
X E (a p) -- (a
0,
xE G-
p2
),
(aP),
where 0 < cl < . . . < G < 1. Then V2 E [0, 1],/2a E F~ ,
/2(x) =
c,,,
x E {e},
x E ( a 2"-' ) -- (a 2" ),
el,
X E (a 2 ) -- (a4),
O,
x E G - (a2),
where 0 < C l < " ' " <¢n < - ' ' < 1 . Then V2E [0,1], /22 E F~ = { {e} . . . . . (a 2" ) . . . . . (a 4), (a 2 ), G}. B y Theorem 1.2(1),/2 E F ( G ) and 1/21 is infinite• This is a contradiction, and shows that our assertion is true. []
= { { e } , ( a P ' ) , ( a p,-~) .... ,(aP~),(aP),G}. B y Theorem 1.2(1), / 2 E F ( G ) and 1/21 > n . This is a contradiction, supueF(G ) 1/21 -- n. Thus 1 < s < n. Case 2: ml > 1, i.e. m = p S m l , p ~ m l . First, we want to prove that s < n - 1. Suppose that s/> n - 1. There is a chain {e} = ( a p°-' ) C (a p"-2) C " " C (a p2) C (a p) C G o f subgroups o f G. Let/2 : G ~ [0, 1] such that Vx E G,
/2(x ) =
Similarly, we can prove:
1,
x E {e},
Cn--1,
X E (a p"-2) -- (a p"-I),
"
C1,
1,0, T h e o r e m 4.2. Let G be a group. Then G is a char-
acter&tically simple group if and only if V/2 E C F ( G ) , 1/21~<2• T h e o r e m 4 . 3 . Let G be a cyclic group and sup•eF(G) 1/21= n . Then either o(G) = pS or o(G)
= pSq, where 1 <~s<~n - 1, both p a n d q are distinct prime and p ~ q. P r o o f . B y T h e o r e m 4.1, we m a y assume G is a finite and G = (a). It is clear that o ( G ) is equal to order o f a . Let order o f a be m. I f m is a prime, then our assertion holds. N o w suppose that m is a c o m p o s i t e number, n a m e l y m = pSml, where both s and ml are positive integers and p is a prime, p ~ m l . This can happen in two cases: Case 1: ml = 1, i.e. m = pS. It is sufficient to prove that 1 < s < n Since m is not a prime, s > 1. Suppose that s/> n. There must exist a chain {e} = (a ps ) C (a ps-' ) C . . . C (a p2 ) C (a p) C G
xE(a p)-(a
p2
),
x E G - (aP),
where 0 < cl < " " < ¢n--1 < 1. T h e n V 2 E [0, 1], /2a E F~= {{e},(aP"-J),(a p"-2) . . . . . (aP2),(aP),G}. By
Theorem 1.2(1),/2 E F ( G ) and IP] > n . This is a contradiction, supu~F(O ) ]/2] = n. Thus 1 < s < n. Second, we want to show that ml is a prime. I f not, then we have ml =nln2, where both nl and n2 are positive integers and nl,n2 > l, n a m e l y m = pSnln2. Consequently, there is a chain {e} =
(a p"-2n' ) C (a p°-2 ) C (a p'-3 ) C...C(aP2)c(aP)c
G
o f subgroups o f G. Let/2 : G ~ [0, 1] such that Vx E G, 1,
/2(x ) =
x c (e},
Cn--1, X E (a p"-2 ) -- (a p"-2n' ), I Cn--2, x C (a p"-3 ) _ (a p.-2 ), C1,
1,0,
xC(a p)-(a x E G - (a p),
p2
),
Y. Zhan 9, K. ZoulFuzzy Sets and Systems 95 (1998) 243-247 w h e r e 0 < c ~ < . . . < c n - ~ < 1 . T h e n \ / 2 E [0, 1],#~ E F u = { { e } , (aP"-~), (aP"-:) . . . . . (aP2),(aP),G}. By Theorem 1.2(1),/a E F ( G ) and I~1 > n . This is a contradiction supueF(6 ) I~1 = n. Thus ml is a prime. This completes the proof. []
247
Proof. Let a ~ b, then
~(xay) = #(xby),
Vx, y E G.
Consequently,
# ( x a b - l y ) = la(xa(b-ly)) =l~(xb(b-ly))=l~(xy),
5. Classification of groups
Vx, y E G ,
Let # be a fuzzy subset o f a group G. For a, b E G, we define that a is equivalent b, written as a ~ b, if and only if
i.e. ab -1 E H . Also, H is a normal subgroup o f G by Theorem 5.1, it follows that a H = bH. Conversely, let a H = bH, then a b - 1 E H. It follows that
#(xay ) = #(xby ),
# ( x a b - l y ) = #(xy),
Vx, y E G.
It is easy to verify that ",-~ " is an equivalent relation on G.
Vx, y E G.
Consequently,
#(xay) = #(x(ab -1 )by) = #(xby),
Vx, y E G.
Theorem 5.1. Let/~ be a f u z z y subset o f a group G, and
Thus, a , - b and this completes the proof.
H = {a E G I p(xay) =- #(xy), Vx, y E G}.
Remark. From Theorem 5.2, a classification o f a group using membership values o f fuzzy subsets is obtained, which is a classification o f cosets about normal subgroup H .
Then H is a normal subgroup o f G. Proof. Va, b E H. Since #(xay) = #(xy),
[]
#(xby) = p(xy),
Vx, y E G,
Acknowledgements we have The authors are very grateful to referees for their advice.
p(xaby ) = #((xa)by) = p(xby ) = l~(xy),
Vx, y E G,
p(xa-l y) = p(x(a-l y) ) = #(xa(a-l y) ) = p(xy),
References
Vx, y E G.
It follows that a, b E H and a - 1 E H . Therefore, H is a subgroup o f G. N o w
p ( x ( z - l a z ) y ) = p ( x z - l z y ) = Iz(xy),
Vx, y E G,
i.e., for VzEG, we have z - l a z E H . normal. []
Thus, H
is
Theorem 5,2. Let ~ be a f u z z y subset o f a group G, and H = {h E G [ l~(xay) =- I~(xy), Vx, y E G}. Then a ~ b if and only if a H = bH f o r Va, b E G.
[1] V.N. Dixit, R. Kumar, N. Ajmal, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990) 359-371. [2] M. Hall Jr., The Theory of Groups, Macmillan, New York, 1959. [3] R. Kumar, Fuzzy characteristic subgroups of a group, Fuzzy Sets and Systems 48 (1992) 397-398. [4] F. Lfiszl6, Structure and construction of fuzzy subgroups of a group, Fuzzy Sets and Systems 51 (1992) 105-109. [5] W. Wangming, Normal fuzzy subgroups, Fuzzy Math. 1 (1981) 21-30. [6] Z. Wenxiu et al., The Introduction of Fuzzy Mathematics, Xian Jiaotong University Publishing House, 1991. [7] Z. Yuanda, Construction of Finite Groups, Science Publishing House, Beijing, 1982.
sets and systems ELSEVIER
Fuzzy Sets and Systems 95 (1998) 243-247
Short Communication
A note on an equivalence relation on fuzzy subgroups 1 Yunjie Zhang*, Kaiqi Zou Department of Basic Science, Dalian Maritime University, Dalian, 116026, People's Republic of China Received May 1996; revised April 1997
Abstract The purpose of this paper is to discuss the following three problems: (1) the number of equivalent classes of fuzzy subgroups of a group G, (2) the relationship between the number of equivalent classes of fuzzy subgroups and the order of a cyclic group G, (3) the relationship between the order of fuzzy subgroups and the order of a cyclic group G, (4) a classification of a group, using membership values of fuzzy subgroups, is given. © 1998 Elsevier Science B.V.
Keywords: Group; Fuzzy subgroup; Equivalent class; Classification
1. Preliminaries
The set of all fuzzy subgroups of a group G will be denoted by F(G).
We recall some definitions and results that will be used later. I f S is a set, then a mapping # : S --* [0, 1] is called a fuzzy subset of S. Let # be a fuzzy subset of S, by an 2-cut #x o f # we mean the following crisp subset of S:
#~= {xlx~S, #(x)>l,~, ,~[o,1]}. Clearly, 2t ~<).2 implies #~, _D#~2. Definition 1.1. Let G be a group. Then a fuzzy subset # of G is said to be a fuzzy subgroup of G if
Definition 1.2. Let G be a group. A # C F(G) is called a normal fuzzy subgroup of G if
g(xyx-1)>...#(y),
Vx, yEG.
Definition 1.3. Let G be a group. A # EF(G) is called a characteristic fuzzy subgroup of G if Vtp E Aut G, g(tp(x)) >~#(x),
Vx E G,
where Aut G is the automorphism group of G.
Theorem 1.1. Let G be a group and #EF(G). Then
#(xy)>~min{~(x),#(y)}, # ( x - l ) / > #(x),
Vx, yEG,
VxEG.
* Corresponding author. I Supported by science foundation of Dalian Maritime University. 0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 1 8 5 - 1
l~ is a normal fuzzy subgroup of G if and only if #(xyx -1) = #(y),
Vx, yEG.
Let G be a group, #EF(G). We shah write Im # for the image set of # and Fu for the family {#~ 12EIm#}.
244
Y. Zhang, K. Zou/Fuzzy Sets and Systems 95 (1998) 243~47
Theorem 1.2. Let G be a group. (1) I f # is a fuzzy subset of G, then I~CF(G) if and
only if V#~ E Fu, #~ ¢ ~, #~ is a subgroup of G. (2) V # E F ( G ) , p is a normal (resp. characteristic) fuzzy subgroup if andonly if Vl~ E F~, I~ ¢ qJ, #~. is a normal (resp. characteristic) subgroup of G (see [3, 5]). (3) I f HoCH1CH2C ... C H r = G is a chain of subgroups, then 3~CF(G) such that F~= {Ho,H1,H2 . . . . . Hr} (see [6, Theorem 10.2.7]). Definition 1.4 (Dixit [1, Definition 3.2]). Let /,,vE F(G). We say that/t is equivalent v, written as/t ,-~ v, ifF u =&.
then I&l =
where
s, = {a},
& = {{G,{e}I,{G,H~} ..... { G , H , _ , }},
&+, = {{e},H1 . . . . . H , _ , , G}. Consequently, rc = ~ = 0 C~ = 2". Case 2: V/-/,.,Hj (i,j ¢O,n), Hi ¢ I-Ij and & g Hi. Let
Ri = {{e},Hi, G},
i= 1,2 . . . . . n - 1
and
Clearly, " ~ " is an equivalent relation on F ( G ) , and F(G) can be classed under the relation " ~ " . The number of equivalent classes will be denoted by re.
Lemma 1.1. L e t G b e a finite group. Then G is a cyclic group if and only if V d l o ( G ), there exists precisely a subgroup of order d (see [7, Theorem 1.4.5 and Corollary]). Definition 1.5 (Ldszl6 [4, Definition 1]). The order of the set of membership values of a fuzzy set/z is called the order o f / t and is denoted by II,I
Theorem 1.3 (Lfiszl6 [4, Corollary 2]). I f in a group G the length of the longest maximal subgroup chain with endpoints {e} and G is n, the order of any fuzzy subgroup of G cannot be greater than n+l.
&={F~C_RilltEF(G) and l14 =k},
k=1,2,3.
Since {G} = S,1 and {{e}, G} cS/2 for Vi (i = 1,2 . . . . . n - 1), from Case I, it follows that rc = ( 2 2 - 2) (n-1)+2=2n. Case 3: We can assume, without loss of generality, that
I{e}
c, = {H,,H2,...,H,,
CH, CH2C.. C2 =
CH,,CG},
{Hn,+,,Hn,+2,...,Hn2 I { e } C H,,+I C . . ' C H,2 C G },
Cm = {mn,n ,+l,Hnm_,+2 . . . . .
O.l{e}
CH,~_,+~ C...CHoC G}, 2. On equivalent classes of fuzzy subgroup Theorem 2.1. Let the number of proper subgroups of a group G be n. Then 2n<~ra <~2". Proof. Let {{e}=Ho,HbH2 ..... Hn=G}
be the set of all subgroups of G. This can happen in three cases: Casel: { e } = H o C H 1 C H 2 C ... CHn = G. From Theorem 1.2(3) and Definition 1.4, we have G C_FI, and l~<]#l~
&--{~ll#l--k},
k=l,2 ..... n+l,
where nl < n 2 < " ' "
Hj gSHk
and
Hk ~ Hj
(k=nm-i + 1,nm-I + 2 . . . . . ni). Similar to Case 1, we have rG = 2 n' + (2 n2
n,+l
_
2) + .-. + (2n-n'-'+l -- 2).
Clearly, 2n ~<2 nl q- ( 2 n 2 - n , + 1 _ 2 ) + - - - + (2 n - n ' - , +~ _ 2) ~<2 n. This completes the proof. []
Y. Zhano, K ZoulFuzzy Sets and Systems 95 (1998) 243-247
Corollary 2.1. I f a group G & a finite, then r6 is finite.
Proof. By Theorem 2.1 and [2, Theorem 1.5.4].
[]
3. ra and cyclic group Theorem 3.1. Let G = ( a ) be a cyclic group and p a prime. (1) I f o ( G ) = p n, then r 6 = 2 n. (2) I f rG = 2 n, then we have either o ( G ) = pn or o ( G ) = pSq, where 1 <~s<~n - 1, both p and q are distinct prime. Proof. (1) Since G = ( a ) be a finite cyclic group, o(G) is equal to order of a. By Lemma 1.1 and Sylow's first theorem (see [2, Theorem 4.2.1]), there must exist a chain {e} :
theorem, there is at least a chain {e} :
Corollary 2.2. A group G is a cyclic group of prime order if and only if re = 2.
(a p") C (a p"-') C " " C (a p2) C (a p) C G
of subgroups of G, a n d {(aP"),(a p" ') . . . . . (ap2), (aP)} is precisely the set of all proper subgroups of G. Consequently, by Theorem 2.1, rG = 2 n. (2) By Corollary 2.1, G = ( a ) is a finite cyclic group, and o(G) is equal to order of a. Let the order of a be m, then we have two cases. Case 1: n = 1. By Corollary 2.2, m is a prime and o(G) = p, where p = m. Case 2: n > 1. By Corollary 2.2, m is not a prime and m = pSq, where p is a prime and PXq, both s and q are positive integers. Case 2.1: q = 1. By Lemma 1.1 and Sylow's first theorem, there exists a chain
245
(a pnq) C (a pn) C (a pn-' ) C " " C (a pz ) C (a p) C G
of subgroups of G. Consequently, rG~>2 n+l by Theorem 2.1 and this contradicts that rc = 2 n. This completes the proof. []
Theorem 3.2. Let G be a group and o( G) = pn. Then G is a cyclic group if and only if re = 2 n. Proof. The necessity is straightforward by Theorem 3.1(1 ). Now, we prove the sufficiency. By Sylow's first theorem, there is a chain {e} =H1 c H 2 c ' " C H n C G of subgroups of G and the order of Hi is pn-i+l ( i = 1 , 2 . . . . . n). Suppose that there exists a subgroup H / ¢ H i and the order of HI is also p,-i+l, then by Theorem 2.1, rG > 2" and this contradicts that r c = 2 n. Therefore, ViC{1,2 . . . . . n}, G has a unique subgroup Hi of order pi. I t follows that G is a cyclic group by Lemma 1.1. With this our proof is complete. [] Let NF(G) (resp. CF(G)) denote the set of all normal (resp. characteristic) fuzzy subgroups of a group G. Then, by Definition 1.4, NF(G) (resp. CF(G)) can be classed under the relation " ~ " . The number of equivalent classes in NF(G) (resp. CF(G)) will be denoted by ru (resp. rc). It is easy to prove that
Theorem 3.3. ( 1) A group G is a simple group if and only if r N = 2. (2) A group G is a characteristically simple group if and only if rc = 2.
{e} =(aP~) c (aP~-' ) C . . . c (aPZ) c (aP) C G
4. I#l and cyclic group of subgroups of G and { ( a P ' ) l i = l , 2 . . . . . s} is precisely the set of all proper subgroups of G. By Theorem 2.1, if s>~n+ 1, then rG~2 n+~ and this contradicts that r6 = 2 n; if s ~
Theorem 4.1. Let G be a cyclic group. Then G & a finite cyclic group if and only if Vp E F(G), [p[ is finite. Proof. The necessity is straightforward by Theorem 1.3. Now, we prove the sufficiency.
Y. Zhano, K. ZoulFuzzy Sets and Systems 95 (1998) 243-247
246
Suppose that G is a infinite cyclic group. Let G --- (a). Then the order o f a is infinite• Consequently, there is a chain
o f subgroups o f G. Let/2 : G ~ [0, 1] such that Vx E G,
{e} C " "
/2(x ) =
C (a 2" ) C ' "
C (a 4) C (a 2 ) C G
o f subgroups o f G. L e t / 2 " G ~ [0, 1] such that Vx E G,
1,
x E {e},
es,
XE (a p~-') -- (a p~),
Cl,
X E (a p) -- (a
0,
xE G-
p2
),
(aP),
where 0 < cl < . . . < G < 1. Then V2 E [0, 1],/2a E F~ ,
/2(x) =
c,,,
x E {e},
x E ( a 2"-' ) -- (a 2" ),
el,
X E (a 2 ) -- (a4),
O,
x E G - (a2),
where 0 < C l < " ' " <¢n < - ' ' < 1 . Then V2E [0,1], /22 E F~ = { {e} . . . . . (a 2" ) . . . . . (a 4), (a 2 ), G}. B y Theorem 1.2(1),/2 E F ( G ) and 1/21 is infinite• This is a contradiction, and shows that our assertion is true. []
= { { e } , ( a P ' ) , ( a p,-~) .... ,(aP~),(aP),G}. B y Theorem 1.2(1), / 2 E F ( G ) and 1/21 > n . This is a contradiction, supueF(G ) 1/21 -- n. Thus 1 < s < n. Case 2: ml > 1, i.e. m = p S m l , p ~ m l . First, we want to prove that s < n - 1. Suppose that s/> n - 1. There is a chain {e} = ( a p°-' ) C (a p"-2) C " " C (a p2) C (a p) C G o f subgroups o f G. Let/2 : G ~ [0, 1] such that Vx E G,
/2(x ) =
Similarly, we can prove:
1,
x E {e},
Cn--1,
X E (a p"-2) -- (a p"-I),
"
C1,
1,0, T h e o r e m 4.2. Let G be a group. Then G is a char-
acter&tically simple group if and only if V/2 E C F ( G ) , 1/21~<2• T h e o r e m 4 . 3 . Let G be a cyclic group and sup•eF(G) 1/21= n . Then either o(G) = pS or o(G)
= pSq, where 1 <~s<~n - 1, both p a n d q are distinct prime and p ~ q. P r o o f . B y T h e o r e m 4.1, we m a y assume G is a finite and G = (a). It is clear that o ( G ) is equal to order o f a . Let order o f a be m. I f m is a prime, then our assertion holds. N o w suppose that m is a c o m p o s i t e number, n a m e l y m = pSml, where both s and ml are positive integers and p is a prime, p ~ m l . This can happen in two cases: Case 1: ml = 1, i.e. m = pS. It is sufficient to prove that 1 < s < n Since m is not a prime, s > 1. Suppose that s/> n. There must exist a chain {e} = (a ps ) C (a ps-' ) C . . . C (a p2 ) C (a p) C G
xE(a p)-(a
p2
),
x E G - (aP),
where 0 < cl < " " < ¢n--1 < 1. T h e n V 2 E [0, 1], /2a E F~= {{e},(aP"-J),(a p"-2) . . . . . (aP2),(aP),G}. By
Theorem 1.2(1),/2 E F ( G ) and IP] > n . This is a contradiction, supu~F(O ) ]/2] = n. Thus 1 < s < n. Second, we want to show that ml is a prime. I f not, then we have ml =nln2, where both nl and n2 are positive integers and nl,n2 > l, n a m e l y m = pSnln2. Consequently, there is a chain {e} =
(a p"-2n' ) C (a p°-2 ) C (a p'-3 ) C...C(aP2)c(aP)c
G
o f subgroups o f G. Let/2 : G ~ [0, 1] such that Vx E G, 1,
/2(x ) =
x c (e},
Cn--1, X E (a p"-2 ) -- (a p"-2n' ), I Cn--2, x C (a p"-3 ) _ (a p.-2 ), C1,
1,0,
xC(a p)-(a x E G - (a p),
p2
),
Y. Zhan 9, K. ZoulFuzzy Sets and Systems 95 (1998) 243-247 w h e r e 0 < c ~ < . . . < c n - ~ < 1 . T h e n \ / 2 E [0, 1],#~ E F u = { { e } , (aP"-~), (aP"-:) . . . . . (aP2),(aP),G}. By Theorem 1.2(1),/a E F ( G ) and I~1 > n . This is a contradiction supueF(6 ) I~1 = n. Thus ml is a prime. This completes the proof. []
247
Proof. Let a ~ b, then
~(xay) = #(xby),
Vx, y E G.
Consequently,
# ( x a b - l y ) = la(xa(b-ly)) =l~(xb(b-ly))=l~(xy),
5. Classification of groups
Vx, y E G ,
Let # be a fuzzy subset o f a group G. For a, b E G, we define that a is equivalent b, written as a ~ b, if and only if
i.e. ab -1 E H . Also, H is a normal subgroup o f G by Theorem 5.1, it follows that a H = bH. Conversely, let a H = bH, then a b - 1 E H. It follows that
#(xay ) = #(xby ),
# ( x a b - l y ) = #(xy),
Vx, y E G.
It is easy to verify that ",-~ " is an equivalent relation on G.
Vx, y E G.
Consequently,
#(xay) = #(x(ab -1 )by) = #(xby),
Vx, y E G.
Theorem 5.1. Let/~ be a f u z z y subset o f a group G, and
Thus, a , - b and this completes the proof.
H = {a E G I p(xay) =- #(xy), Vx, y E G}.
Remark. From Theorem 5.2, a classification o f a group using membership values o f fuzzy subsets is obtained, which is a classification o f cosets about normal subgroup H .
Then H is a normal subgroup o f G. Proof. Va, b E H. Since #(xay) = #(xy),
[]
#(xby) = p(xy),
Vx, y E G,
Acknowledgements we have The authors are very grateful to referees for their advice.
p(xaby ) = #((xa)by) = p(xby ) = l~(xy),
Vx, y E G,
p(xa-l y) = p(x(a-l y) ) = #(xa(a-l y) ) = p(xy),
References
Vx, y E G.
It follows that a, b E H and a - 1 E H . Therefore, H is a subgroup o f G. N o w
p ( x ( z - l a z ) y ) = p ( x z - l z y ) = Iz(xy),
Vx, y E G,
i.e., for VzEG, we have z - l a z E H . normal. []
Thus, H
is
Theorem 5,2. Let ~ be a f u z z y subset o f a group G, and H = {h E G [ l~(xay) =- I~(xy), Vx, y E G}. Then a ~ b if and only if a H = bH f o r Va, b E G.
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