Fuzzy subgroups having the property (∗) II

Fuzzy Sets and Systems 64 (1994) 245-248 North-Holland

245

Fuzzy subgroups having the property (,) II Jae-Gyeom

Kim

Department of Mathematics, Kyungsung University, Pusan 608-736, South Korea Received October 1993 Revised February 1994

Abstract: In [3], the notion of the property (*) of a fuzzy subgroup of a group, which is a more fuzzified notion of the notion of a generalized characteristic fuzzy subgroup of a finite group, is introduced. In this paper, we show that a generalized characteristic fuzzy subgroup of any group has always the property (*). Keywords: Fuzzy subgroup; generalized characteristic fuzzy subgroup; the property (*); fully invariant fuzzy subgroup.

1. Introduction Abou-Zaid [1] introduced the notion of a generalized characteristic fuzzy subgroup of a finite group and gave a characterization of all finite cyclic groups in terms of generalized characteristic fuzzy subgroups. In [3], using the notion of fuzzy orders of elements, the author introduced the notion of the property (*) of a fuzzy subgroup of a group, which is a more fuzzified notion of the notion proposed by Abou-Zaid, showed that a generalized characteristic fuzzy subgroup of a finite nilpotent group has the property (*), and characterized all finite cyclic groups in terms of this more fuzzified notion. And in [3, 5], the author investigated relationships between the property (*) and some properties of a fuzzy subgroup. In this paper, we apply the notion of a generalized characteristic fuzzy subgroup of a finite group to any group, show that a generalized characteristic fuzzy subgroup of any group has always the property (*), and investigate some conditions for a fuzzy subgroup having the property (*) to be a generalized Correspondence to: Dr. J.-G. Kim, Department of Mathematics, Kyungsung University, Pusan, 608-736 South Korea.

characteristic fuzzy subgroup or to be a fully invariant fuzzy subgroup. Throughout this paper, we shall denote the identity of a group G by e, the order of x in G by O(x), and the greatest common divisor of integers m and n by (m, n). 2. The property (*) Definition 2.1 [4]. Let A be a fuzzy subgroup of a group G. For a given x in G, the least positive integer n such that A(x n) = A ( e ) is called the fuzzy order of x with respect to A (briefly, FOA(x)). If no such such n exists, x is said to have the infinite fuzzy order with respect to A. In that case, we write FOA(X) = oo. FOA(x) is always a divisor of O(x) [4]. If GA={e}, then F O A ( x ) = O ( x ) for all x e G , where GA denotes the set {x e G [A(x) =A(e)}. However, the two statements O(x)= O(y) and F O A ( x ) = F O A ( y ) are independent of each other, as is illustrated in the following example.

Example 2.2. Define a fuzzy subgroup A of 7/2 x 7/4 by A(x) = to if x e ((0, 2)) and A(x) = t~ otherwise, where to>t1. Then FOA((0, 1))= FOA((1, 0)) = 2, but O((0, 1)) = 4 and O((1, 0)) = 2. And O((1, 0)) -- O((0, 2)) = 2, but FOA((1, 0)) = 2 and FOA((0, 2)) = 1. Definition 2.3 [cf. 1]. A fuzzy subgroup A of a group G is called an (extended) generalized characteristic fuzzy subgroup (briefly, a GCFS) of G if O(x) = O(y) implies A(x) = A(y) for all x, y i n G .

Remark 2.4. In [1], the notion of a GCFS was defined for a finite group. So the notion of a GCFS in Definition 2.3 is an application of the old notion. And all results in [1] are valid where old GCFSs are replaced by new GCFSs, in particular, a GCFS of a group is a fuzzy characteristic subgroup of the group [1, 6] and so a normal fuzzy subgroup of the group [7].

0165o0114/94/$07.00 ~) 1994--Elsevier Science B.V. All rights reserved SSDI 0165-0114(94)00085-L

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J.-G. Kim / Fuzzy subgroups having the property (*) H

O(x) is a nonfuzzy notion and the notion

FOA(X ) is a fuzzified notion of O(x). So the notion of the property (*) of a fuzzy subgroup in the following definition is a more fuzzified notion of the notion of a GCFS.

Definition 2.5 [3]. A fuzzy subgroup A of a group G is said to have the property (*) if FOA(x) = FOA(y) implies A ( x ) -- A ( y ) for all x, y i n G. Theorem 2.6 [2]. Let A be a fuzzy subgroup of a group G. For all x in G, the following hold: (1) I f A ( X m) = A(e) for some positive integer m, then FOa(x) divides m. (2) (FOA(X), m ) = 1 for some positive integer m, then A ( x m) = A(x). (3) FOA(Xm) = F O A ( X ) / ( F O A ( X ) , m ) for all positive integers m.

by T h e o r e m 2.6(3), and so

A (x p~') = A (yP~') by the induction hypothesis, for each i e {1 . . . . . r}. And there exist integers cl . . . . . cr such that 1 = clp~' + • • • + crp~" because p~' . . . . . pT" are coprime. Thus we have

A ( x ) = A ( x c'p~'÷'''÷c'p:r) ~> min{A(xC,p~J). . . .

, A(x~'e~')}

>- min{A(xP~ ') . . . . .

A(xP'~')}

and so

A ( x ) = min{A(x pg'). . . . .

A(xP:r)}.

Similarly, we have

A ( y ) = min{A(yP~'), . . . , A(yP'~')}. Hence we have A ( x ) = A ( y ) because

Theorem 2.7. Let A be a GCFS of a finite group G. Then A has the property (*).

A (xP';') -_ A(yPT')

Proof. Let x , y e G such that FOA(X)= F O A ( y ) = n . If n = l , then A ( x ) = A ( y ) obviously. Thus let n = p~' • • • pT' > 1, where Pi are distinct primes and 0~'i --i 1 for all i e {1 . . . . , r}. We now use induction on r. Let r = 1. By T h e o r e m 2.6(1), O ( x ) = na and O ( y ) = n b for some positive integers a and b, and so O(x) =n(a, b)s and O ( y ) = n ( a , b)t for some positive integers s and t with (s, t ) = 1. H e r e , either (n, s) = 1 or (n, t) = 1 because (s, t) = 1 and r = l , so we may assume that ( n , s ) = l . Since A is a GCFS and O ( ( y ' ) " ) = O((x')m), A((yt) m) = A((xs)ra), for all positive integers m. Thus FOA(y') = FOA(x~). But

Lemma 2,8. Let A be a f u z z y subgroup of a group G such that the following property holds: (a) If A ( x ) = A ( e ) and O ( x ) = O ( y ) , then A ( y ) = A(e), for all x, y e G. If O(x) = ~, then FOA(x) = 1 or ~, for all x in G.

FOA(X ~) = FOA(X)/(FOA(X), s)

= n/(n, s) = n = FOA(y ) by T h e o r e m 2.6(3). Thus F O A ( y ' ) = F O a ( y ) , and so (FOA(y), t) = 1 by T h e o r e m 2.6(3) again. Therefore A ( y ) = A ( y ' ) by T h e o r e m 2.6(2). And, since O ( y ' ) = O ( y ~) and ( n , s ) = l , A ( y ' ) = A ( x ~) = A ( x ) by T h e o r e m 2.6(2) and the assumption that A is a GCFS. Hence we have A ( x ) = A ( y ) . Now let r > 1. Then

FOA (Xp~') ----FOA (yP:') _-p~, . . . . . . . . . . . ,... Pi-lPi+l P7 ~

for all i e {1 . . . . .

r}. This completes the proof.

Proof. Let x e G such that O ( x ) = o o and FOA(X) = n < ~. Then A(x") = A(e). But O(x) = O(x") = o0. So A ( x ) = A(e) by property (a). This completes the proof. Corollary 2.9. Let A be a GCFS o]" a group G. Then A has the property (*). Proof. Let x, y e G such that FOA(X) = FOA(y) = n. If n = 1, then A(x) = A ( y ) = A(e) obviously. If n =o0, then O ( x ) = O ( y ) = obviously, and so A ( x ) - - A ( y ) because A is a GCFS. And if 1 < n < 0% then O(x), O(y) < ~, by L e m m a 2.8, because a CGFS A has the property (a) in L e m m a 2.8, and so A ( x ) - - A ( y ) by the proof of T h e o r e m 2.7. So the notion of the property (*) of a fuzzy subgroup is a generalized and more fuzzified notion of the notion of a GCFS.

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3. Relationships between the property (*) and other properties

of G if the image f ( A ) of A under f is contained in A for all endomorphisms f of G.

Proposition 3.1. Let A be a f u z z y subgroup of a group G such that GA is normal in G. A n d let :r be a natural group homomorphism from G onto G/GA. Then A has the property (*) if and only if the image st(A) of A under Jr is a GCFS of G/GA.

Proposition 3.5 [8]. Let f be a group homomorphism from G into H. Let A and B be f u z z y subgroups of G and H, respectively. Then f ( A ) ~_ B if and only if B ( f ( x ) ) >>-A(x) for all x inG.

Proof. For all x ~ G, we have FOa(x) = O(ar(x)) obviously.

Lemma 3.2. Let A be a fuzzy subgroup of a group G having property (a) in Lemma 2.8. Then O(x) = O ( y ) implies FOA(X) = FOm(y) for all x, y ~ G. Proof. Let x, y ~ G such that O(x) = O(y) = n. If n = oo, then we have FOA(x) = 1 ¢:~ A ( x ) = A(e)

¢:> A ( y ) = A(e) (by the property (a))

Proposition 3.6. Let A be a GCFS of a group G such that the following property holds: (b) If O ( x ) < ~ and O ( y ) = ~, then A(x)>1 A(y), for all x, y ~ G. Then A is a fully invariant f u z z y sugbroup of G. Proof. First, if O ( y ) divides O(x) for some x, y e G with O(x) ~ A ( x ) because A is a GCFS. Next, let f be an endomorphism of G. If O ( x ) < oo, then O ( f ( x ) ) divides O(x), and so A ( f ( x ) ) > - A ( x ) . And if O(x) = o~, then A ( f ( x ) ) >~A(x) because a GCFS A has property (b). Thus A is a fully invariant fuzzy subgroup of G by Proposition 3.5.

¢::> FOm(y) = 1, and so FOA(x)= FOA(y) by Lemma 2.8. Thus let n < ~, and let FOA(X)=s and FOm(y)= t. Then A ( x s ) = A ( e ) , and so A ( y s ) = A ( e ) , by property (a), because O(xS)=O(yS). Thus t divides s by Theorem 2.6(1). Similarly, s divides t. Hence we have FOA(X)=FOA(y). By Lemma 3.2, we have a theorem obviously.

Theorem 3.3. Let A be a f u z z y subgroup of a group G having the property (*). Then A is a GCFS of G if and only if A has property (a) in Lemma 2.8. So GA completely determines whether a fuzzy subgroup A of a group G having the property (*) is a GCFS. And GA completely determines other properties of the fuzzy subgroup A, in fact, A is a normal fuzzy subgroup or a fuzzy characteristic subgroup of G if and only if GA is normal or characteristic in G, respectively [5].

Definition 3.4 [8]. A fuzzy subgroup A of a group G is called a fully invariant fuzzy subgroup

Corollary 3.7. Let A be a fuzzy subgroup of a group G having the property (*), property (a) in Lemma 2.8, and property (b) in Proposition 3.6. Then A is a fully invariant fuzzy subgroup of G. In Proposition 3.6 and Corollary 3.7, the assumption that A has property (b) cannot be omitted (see Example 3.8). In Corollary 3.7, the assumption that A has property (a) cannot be omitted (see Example 3.9). And the converses of Proposition 3.6 and Corollary 3.7 do not hold in general (see Example 3.10).

Example 3.8. Define a fuzzy subgroup A of a group G = ( a , b l a 2 = b 2 = e ) by A ( e ) = t o , A ( x ) = tl if x ~ ( a b ) \ { e } , and A(x) = t2 otherwise, where to > tl > t2. Then A is a GCFS of G and has the property (*) and property (a), but A does not have property (b). Define an endomorphism f of G by f ( x ) = e if the number of a's in an expression for x is even and f ( x ) = a otherwise. Then A ( f ( a b ) ) = A(a) = t2 < tl = A(ab). So A is not a fully invariant fuzzy subgroup of G by Proposition 3.5.

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E x a m p l e 3.9. Define a fuzzy subgroup A of 7/2 x Z4 by A ( X ) = to if x e ((1, 2)) and A(x) = tl otherwise, where t o > t l . Then A has the property (*) and the property (b), but A does not have the property (a). Define an endomorphism f of Z 2 x Z 4 by f((a, b ) ) = (0, b). Then A ( f ( ( l , 2))) -- A((0, 2)) = tl < to = A((1, 2)). So A is not a fully invariant fuzzy subgroup of G by Proposition 3.5.

Example 3.10. Define a fuzzy subgroup A of Z2 x Z4 by A ( e ) = to, A((O, 2)) = tl, and A ( x ) = t2 if x $ ((0, 2)), where to > tl > t2. Then A is a fully invariant fuzzy subgroup of G. But A has not the property (*) and is not a GCFS of G. I.emma 3.11. Let A be a GCFS of an Abelian group G. Then A has the property (b) in Proposition 3.6. Proof. Let x , y E G such that O ( x ) < o o and O(y) = oo. Then O(xy) = oo. So A ( y ) = a ( x y ) because A is a GCFS. Thus we have

A(x) =A((xy)(y-1)) >! min{A(xy), A ( y - 1 ) } = A ( y ) .

Corollary 3.12. Let A be a GCFS of an Abelian group G or a fuzzy subgroup of G having the property (*) and property (a) in Lemma 2.8. Then A is a fully invariant fuzzy subgroup of G.

References

[1] S. Abou-Zaid, On generalized characteristic subgroups of a finite group, Fuzzy Sets and Systems 43 (1991) 235-241. [2] J.G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci. (to appear). [3] J.G. Kim, Fuzzy subgroups having the property (*), Inform. Sci. (to appear). [4] J.G. Kim, On fuzzy orders of elements of fuzzy subgroups, J. Kyungsung Univ. 13 (1992) 251-257. [5] J.G. Kim, Some properties of fuzzy subgroups, submitted. [6] R. Kumar, Fuzzy characteristic subgroups of a group, Fuzzy Sets and Systems 48 (1992) 397-398. [7] N.P. Mukherjee and P. Bhattacharya, Fuzzy groups: Some group-theoretic analogs, Inform. Sci. 39 (1986) 247-268. [8] F.I. Sidky and M.A.A. Mishref, Fuzzy invariant, characteristic, and S-fuzzy subgroups, Inform. Sci. 55 (1991) 27-33.