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Groups and fuzzy subgroups II
Fuzzy Sets and Systems 56 (1993) 375-377 North-Holland
375
Short Communication
Groups and fuzzy subgroups II Mohamed Asaad and Salah Abou-Zaid Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt
Definition 1.4. L e t A be a fuzzy subset of S. F o r t e [0, 1], the subset A t =- {X E S I A ( x ) ~ t} is called a level subset of the fuzzy subset A.
Received July 1992 Revised November 1992
W e n e e d the following well-known results:
Abstract: In 1991, Asaad has suggested the following
T h e o r e m 1.5 [2; T h e o r e m 3.2.]. A n y subgroup H o f a group G can be realised as a level subgroup o f some f u z z y subgroup o f G.
question: What can be said about the structure of a group G when information is known about the structure of a fuzzy subgroup A of G and vice-versa?. The purpose of this paper is to continue the investigation of the above mentioned question.
Keywords: Fuzzy subgroup; fuzzy normal subgroup; level subsets of the fuzzy subset; Sylow subgroup; 7r-Hall subgroup.
T h e o r e m 1.6 [(4; T h e o r e m 3.6, 3.9), (2; T h e o r e m 2.1, 2.2)]. Let G be a group and A be a f u z z y subset o f G. Then A is a f u z z y (normal) subgroup o f G if and only if the level subsets At, for t ~ [0, 1], t <~A ( e ) are (normal) subgroups o f G.
1. Preliminaries
Definition 1.1. L e t S be a set. A fuzzy subset A of S is a m a p A:S--~ [0, 1].
T h e o r e m 1.7 ( D e d e k i n d T h e o r e m ) . Let G be a non-Abelian finite group o f which all proper subgroups are normal Then we have G = Q x C x E where Q is the quaternion group o f order 8, C is an Abelian group in which every element is o f odd order and E is an Abelian group o f exponent 2 or 1.
Definition 1.2. L e t G be a group. A fuzzy subset A of G is said to be a fuzzy s u b g r o u p of G if (i) A ( x y ) ~ min(A(x), A ( y ) ) for all x, y in G, (ii) A ( x ) = A ( x -1) for all x in G.
Definition 1.8. A ~r-Hall s u b g r o u p of G is a n - s u b g r o u p of G whose index in G is not divisible by any prime in Jr. A Sylow p - s u b g r o u p of G is a m a x i m a l p - s u b g r o u p of G.
W e first recall s o m e basic definitions for the sake of completeness.
Definition 1.3. L e t G be a group. A fuzzy subgroup A of G is called n o r m a l if A ( x ) = A ( y x y - 1 ) , for all x, y in G.
Correspondence to: Dr. S. Abou-Zaid, Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt.
2. On Abelian finite groups First we recall that if G is any g r o u p and is usually d e n o t e d by Ix, y] and is called the c o m m u t a t o r o f x and y. W e n e e d the following two lemmas.
x , y e G, then x - l y - l x y
0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved
376
M. Asaad, S. Abou-Zaid / Groups and fuzzy subgroups II
Lemma 2.1 [5; Proposition 5.5]. Let G be a group and A be a f u z z y subgroup o f G. I f A ( x y -1) = A(e), then A ( x ) = A ( y ) . Lemma 2.2. Let G be a group and A be a f u z z y subgroup o f G. I f A ( [ x , y ] ) = A ( e ) for all x, y • G, then A is normal. Proof. Let A ( x - l y - l x y ) = A ( e ) for all x , y in G then, by applying L e m m a 2.1, we get A ( y - ~ x y ) = A ( x ) and hence A is fuzzy normal.
that any subgroup of G is normal. Now by applying the Dedekind T h e o r e m 1.7, we have G = Q × C x E where Q is the quaternion group of order 8, C is an Abelian group in which every element is of odd order and E is an Abelian group of exponent 2 or 1. Since each subgroup of G is Abelian and Q is not Abelian, we must have E = C = (e) and so G = Q. This is impossible, as there exists a fuzzy subgroup A of Q which does not satisfy the condition. For instance, consider the fuzzy subgroup A: A : Q = (a, b ] a 4 = e, a 2 = b 2, ba
The converse of L e m m a 2.2 is not true in general, for example, consider G = $3 = {a, b I a2 = b 3 -= e, ab 2 = ba}. Define a fuzzy subgroup A as follows:
A(x) =
X • {b, b2}, otherwise.
Since the level subgroups of A are normal by T h e o r e m 1.6, A is a fuzzy normal subgroup. But A does not satisfy the condition of L e m m a 2.2 as A ( a - l b - l a b ) : A(b 2) = ~ ~ A(e). We now prove the next theorem. Theorem 2.3. Let G be a finite group. Then G is Abelian if and only if for any f u z z y subgroup A o f G, we have A([x, y]) = A ( e ) for all x, y • G. Proof. First we assume G is Abelian. Then for all x, y in G, [x, y] = e and hence A([x, y]) = A ( e ) for any fuzzy subgroup A of G. Conversely, we assume that any fuzzy subgroup A of G satisfies the condition A([x, y]) = A ( e ) for all x, y in G. Now we show that G is Abelian. Let H be any proper subgroup of G and A be any fuzzy subgroup of H. Clearly, by defining an extension map A0 of A such that Ao(x) = 0 for all x in G \ H , it follows easily that A0 is a fuzzy subgroup of G and hence by hypothesis Ao satisfies the condition Ao([X, y]) = Ao(e) for all x, y in G. Also it is clear that the restriction of A0 to H (which is A) satisfies such condition for all x, y in H. Then H is Abelian by induction on the order of G, and hence any proper subgroup of G is Abelian. L e m m a 2.2 implies that any fuzzy subgroup of G is normal. Then apply T h e o r e m 1.6 and we have
A(x)=
1
x~--e~
tt
X
t2
otherwise,
= a-Xb)-----~
[0,
1],
a 2, 1 > t I > t 2.
We note that A([a, b]) = A(a 2) = tl # A ( e ) . Therefore G is Abelian and the proof is complete.
3. On the existance of normal Hall subgroups
In this section we introduce the following: Let G be a finite group such that O ( G ) is divisible by at least two distinct primes. Let p be a prime divisor of O(G). We define the two sets H and K as follows: H = {x • G [ x is a non-trivial p-element of G}, K = {x • G [ x is a non-trivial p ' - e l e m e n t of G}. Let A be a fuzzy subset of G such that t I = min{A(x) lx EH} and t2 = max{A(x) lx e g } . Now we prove this theorem.
Theorem 3.1. Let G be a group o f order div&ible by at least two distinct primes. Let P be a Sylow p-subgroup o f G. Then P is normal in G if and only if there exists a f u z z y subgroup A o f G such that tl > t2.
Proof. Suppose that P is a normal Sylow p-subgroup of G. Then P = {e} U H where H is defined as above. We define a fuzzy subset A of
M. Asaad, S. Abou-Zaid / Groups and fuzzy subgroups II G as follows:
A ( x ) = {tl if x cP, t2 i f x ~ p , where 0 ~< t2 < t~ ~< 1. Then if follows easily, by T h e o r e m 1.6, that A is a fuzzy subgroup of G. Also we have min{A(x) Ix e l l } = min{A(x) ] x E P} = t~ > t2 = max{A(x) ] x ~ P} = max{A(x) I x EK}. Conversely, suppose that A is a fuzzy subgroup of G such that tl > t2. Then A,1 = {x E G I A ( x ) >i t~} is a level subgroup of A. We prove that A,, is a p-subgroup of G. Since each element of P is a p-element, we have that P is a subgroup of A,1. Since G is of order divisible by at least 2-distinct primes, then there exists a prime q ( ¢ p ) such that q [ O(G). By Cauchy's T h e o r e m II.5.2. of [3], G contains an element y of order q. We prove that y eAt,. If not Y ~ A t l , then A ( y ) ~ t l > t z = m a x { A ( x ) [ x E K } >~A(y) and this is impossible. Thus q Jf O(At,) for any prime q ( ~ p ) . So At, is a p-subgroup of G. Since P ~_ A,1 and P is a Sylow p-subgroup of A,,, we have P = Ate. Now it is not difficult to show that A,, is a normal subgroup of G. Since
377
O(x) = O ( y - l x y ) for all x, y in G then if x cA,, we have y - l x y eAt1 for all y in G. This finishes the proof of the theorem. As a corollary of the proof of T h e o r e m 3.1, we have: Corollary 3.2. Let L be a proper re-Hall subgroup o f G. Then L is a normal subgroup in G if and only if there exists a f u z z y subgroup A o f G such that tl > t2, where tl = min{A(x) I x is a non-trivial 7r-element of G} and t2 = max{A(x) ] x is a non-trivial zc'-element of G}.
References [1] M. Asaad, Groups and fuzzy subgroups, Fuzzy Sets and Systems 39 (1991) 323-328. [2] P.S. Das, Fuzzy groups and level subgroups, J. Math. AnaL Appl. 84 (1981) 264-269. [3] T.W. Hungerford, Algebra (Springer-Verlag, New York, 1974). [4] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [5] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517.
375
Short Communication
Groups and fuzzy subgroups II Mohamed Asaad and Salah Abou-Zaid Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt
Definition 1.4. L e t A be a fuzzy subset of S. F o r t e [0, 1], the subset A t =- {X E S I A ( x ) ~ t} is called a level subset of the fuzzy subset A.
Received July 1992 Revised November 1992
W e n e e d the following well-known results:
Abstract: In 1991, Asaad has suggested the following
T h e o r e m 1.5 [2; T h e o r e m 3.2.]. A n y subgroup H o f a group G can be realised as a level subgroup o f some f u z z y subgroup o f G.
question: What can be said about the structure of a group G when information is known about the structure of a fuzzy subgroup A of G and vice-versa?. The purpose of this paper is to continue the investigation of the above mentioned question.
Keywords: Fuzzy subgroup; fuzzy normal subgroup; level subsets of the fuzzy subset; Sylow subgroup; 7r-Hall subgroup.
T h e o r e m 1.6 [(4; T h e o r e m 3.6, 3.9), (2; T h e o r e m 2.1, 2.2)]. Let G be a group and A be a f u z z y subset o f G. Then A is a f u z z y (normal) subgroup o f G if and only if the level subsets At, for t ~ [0, 1], t <~A ( e ) are (normal) subgroups o f G.
1. Preliminaries
Definition 1.1. L e t S be a set. A fuzzy subset A of S is a m a p A:S--~ [0, 1].
T h e o r e m 1.7 ( D e d e k i n d T h e o r e m ) . Let G be a non-Abelian finite group o f which all proper subgroups are normal Then we have G = Q x C x E where Q is the quaternion group o f order 8, C is an Abelian group in which every element is o f odd order and E is an Abelian group o f exponent 2 or 1.
Definition 1.2. L e t G be a group. A fuzzy subset A of G is said to be a fuzzy s u b g r o u p of G if (i) A ( x y ) ~ min(A(x), A ( y ) ) for all x, y in G, (ii) A ( x ) = A ( x -1) for all x in G.
Definition 1.8. A ~r-Hall s u b g r o u p of G is a n - s u b g r o u p of G whose index in G is not divisible by any prime in Jr. A Sylow p - s u b g r o u p of G is a m a x i m a l p - s u b g r o u p of G.
W e first recall s o m e basic definitions for the sake of completeness.
Definition 1.3. L e t G be a group. A fuzzy subgroup A of G is called n o r m a l if A ( x ) = A ( y x y - 1 ) , for all x, y in G.
Correspondence to: Dr. S. Abou-Zaid, Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt.
2. On Abelian finite groups First we recall that if G is any g r o u p and is usually d e n o t e d by Ix, y] and is called the c o m m u t a t o r o f x and y. W e n e e d the following two lemmas.
x , y e G, then x - l y - l x y
0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved
376
M. Asaad, S. Abou-Zaid / Groups and fuzzy subgroups II
Lemma 2.1 [5; Proposition 5.5]. Let G be a group and A be a f u z z y subgroup o f G. I f A ( x y -1) = A(e), then A ( x ) = A ( y ) . Lemma 2.2. Let G be a group and A be a f u z z y subgroup o f G. I f A ( [ x , y ] ) = A ( e ) for all x, y • G, then A is normal. Proof. Let A ( x - l y - l x y ) = A ( e ) for all x , y in G then, by applying L e m m a 2.1, we get A ( y - ~ x y ) = A ( x ) and hence A is fuzzy normal.
that any subgroup of G is normal. Now by applying the Dedekind T h e o r e m 1.7, we have G = Q × C x E where Q is the quaternion group of order 8, C is an Abelian group in which every element is of odd order and E is an Abelian group of exponent 2 or 1. Since each subgroup of G is Abelian and Q is not Abelian, we must have E = C = (e) and so G = Q. This is impossible, as there exists a fuzzy subgroup A of Q which does not satisfy the condition. For instance, consider the fuzzy subgroup A: A : Q = (a, b ] a 4 = e, a 2 = b 2, ba
The converse of L e m m a 2.2 is not true in general, for example, consider G = $3 = {a, b I a2 = b 3 -= e, ab 2 = ba}. Define a fuzzy subgroup A as follows:
A(x) =
X • {b, b2}, otherwise.
Since the level subgroups of A are normal by T h e o r e m 1.6, A is a fuzzy normal subgroup. But A does not satisfy the condition of L e m m a 2.2 as A ( a - l b - l a b ) : A(b 2) = ~ ~ A(e). We now prove the next theorem. Theorem 2.3. Let G be a finite group. Then G is Abelian if and only if for any f u z z y subgroup A o f G, we have A([x, y]) = A ( e ) for all x, y • G. Proof. First we assume G is Abelian. Then for all x, y in G, [x, y] = e and hence A([x, y]) = A ( e ) for any fuzzy subgroup A of G. Conversely, we assume that any fuzzy subgroup A of G satisfies the condition A([x, y]) = A ( e ) for all x, y in G. Now we show that G is Abelian. Let H be any proper subgroup of G and A be any fuzzy subgroup of H. Clearly, by defining an extension map A0 of A such that Ao(x) = 0 for all x in G \ H , it follows easily that A0 is a fuzzy subgroup of G and hence by hypothesis Ao satisfies the condition Ao([X, y]) = Ao(e) for all x, y in G. Also it is clear that the restriction of A0 to H (which is A) satisfies such condition for all x, y in H. Then H is Abelian by induction on the order of G, and hence any proper subgroup of G is Abelian. L e m m a 2.2 implies that any fuzzy subgroup of G is normal. Then apply T h e o r e m 1.6 and we have
A(x)=
1
x~--e~
tt
X
t2
otherwise,
= a-Xb)-----~
[0,
1],
a 2, 1 > t I > t 2.
We note that A([a, b]) = A(a 2) = tl # A ( e ) . Therefore G is Abelian and the proof is complete.
3. On the existance of normal Hall subgroups
In this section we introduce the following: Let G be a finite group such that O ( G ) is divisible by at least two distinct primes. Let p be a prime divisor of O(G). We define the two sets H and K as follows: H = {x • G [ x is a non-trivial p-element of G}, K = {x • G [ x is a non-trivial p ' - e l e m e n t of G}. Let A be a fuzzy subset of G such that t I = min{A(x) lx EH} and t2 = max{A(x) lx e g } . Now we prove this theorem.
Theorem 3.1. Let G be a group o f order div&ible by at least two distinct primes. Let P be a Sylow p-subgroup o f G. Then P is normal in G if and only if there exists a f u z z y subgroup A o f G such that tl > t2.
Proof. Suppose that P is a normal Sylow p-subgroup of G. Then P = {e} U H where H is defined as above. We define a fuzzy subset A of
M. Asaad, S. Abou-Zaid / Groups and fuzzy subgroups II G as follows:
A ( x ) = {tl if x cP, t2 i f x ~ p , where 0 ~< t2 < t~ ~< 1. Then if follows easily, by T h e o r e m 1.6, that A is a fuzzy subgroup of G. Also we have min{A(x) Ix e l l } = min{A(x) ] x E P} = t~ > t2 = max{A(x) ] x ~ P} = max{A(x) I x EK}. Conversely, suppose that A is a fuzzy subgroup of G such that tl > t2. Then A,1 = {x E G I A ( x ) >i t~} is a level subgroup of A. We prove that A,, is a p-subgroup of G. Since each element of P is a p-element, we have that P is a subgroup of A,1. Since G is of order divisible by at least 2-distinct primes, then there exists a prime q ( ¢ p ) such that q [ O(G). By Cauchy's T h e o r e m II.5.2. of [3], G contains an element y of order q. We prove that y eAt,. If not Y ~ A t l , then A ( y ) ~ t l > t z = m a x { A ( x ) [ x E K } >~A(y) and this is impossible. Thus q Jf O(At,) for any prime q ( ~ p ) . So At, is a p-subgroup of G. Since P ~_ A,1 and P is a Sylow p-subgroup of A,,, we have P = Ate. Now it is not difficult to show that A,, is a normal subgroup of G. Since
377
O(x) = O ( y - l x y ) for all x, y in G then if x cA,, we have y - l x y eAt1 for all y in G. This finishes the proof of the theorem. As a corollary of the proof of T h e o r e m 3.1, we have: Corollary 3.2. Let L be a proper re-Hall subgroup o f G. Then L is a normal subgroup in G if and only if there exists a f u z z y subgroup A o f G such that tl > t2, where tl = min{A(x) I x is a non-trivial 7r-element of G} and t2 = max{A(x) ] x is a non-trivial zc'-element of G}.
References [1] M. Asaad, Groups and fuzzy subgroups, Fuzzy Sets and Systems 39 (1991) 323-328. [2] P.S. Das, Fuzzy groups and level subgroups, J. Math. AnaL Appl. 84 (1981) 264-269. [3] T.W. Hungerford, Algebra (Springer-Verlag, New York, 1974). [4] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [5] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517.