- Email: [email protected]
Fuzzy groups: Some group theoretic analogs. II
INFORMATION
SCIENCES
41, 77-91 (1987)
77
Fuzzy Groups Some Group ‘Ilteoretic Analogs. II PRABIR BHATTACHARYA
*
Depurtment of Muthemntics, Kuwait University, P.O. Box 5969, 13060 S&at, Kuwuit and N. P. MUKHERJEE School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi 110067, India Communicated
by Azriel Rosenfeld
ABSTRACT We introduce the concepts of (1) order of a fuzzy subgroup of a finite group, (2) a fuzzy Abelian group, and (3) a fuzzy solvable group. We obtain some results involving these notions which are analogs of results from group theory. Also, we prove analogs of two basic theorems of group theory, namely Cayley’s theorem and Lagrange’s theorem.
1.
INTRODUCTION
The present paper is a sequel to [9]. The concept of a fuzzy group was introduced by Rosenfeld in (lo]; he obtained a number of analogs of results from group theory. There have been some further ~vestigations to characterize fuzzy groups; see for example [l-2], [6], [7]. Through a series of papers [3-5, 8-91 we have attempted to investigate fuzzy groups and obtained several interesting characterizations. In particular we have obtained a number of fuzzy analogs of concepts from (finite) group theory and proved the fuzzy versions of some important group theoretical properties. Group theory plays a prominent role in mathematical sciences with a number of practical applications in areas like particle physics, coding theory and combinatorics, to name just a few. Our investigations have been motivated by interest in the error correcting capabilities of a linear code over a finite field. We obtain in the present paper the fuzzy analogs of two classical results from group theory, namely Cayley’s theorem and Lagrange’s theorem. We develop here the notions of the “order” of a fuzzy group and fuzzy Abelian group. Further we introduce the concept of a fuzzy solvable group. We obtain some analogs of group theoretical results related to these ideas introduced here. Our primary aim has been to indicate that fuzzy groups have a rich structure which *Present address: Department of Computer Science, University of Nebraska-Lincoln, Ferguson Bldg., Lincoln, NE 68588-0115. OElsevier Science Publishing Co., Inc. 1987 52 Vanderbilt Ave., New York, NY 10017
m20-0255/87/$03.50
PRABIR BHA’ITACHARYA
78
AND N. P. MUKHERJEE
is worth exploring, and some of the properties of a fuzzy group (but not all) are nice analogs of group theoretic results.
2.
PRELIMINARIES
We briefly review here some definitions Rosenfeld [lo] and our papers [3-51, [8-91.
and results. For more details see
DEFINITION [lo]. Let G be a group. A map p : G + [0, l] is called a fuzzy subgroup if (i) P(V) 2 ~NP(~),P(Y)) (ii) /.Qx-‘) = p(x) Vx E G.
vx, Y E G,
DEFINEION [8]. Let p be a fuzzy subgroup fuzzy normal if I = I Vx, y E G.
on a group G. Then ~1is called
DEFINITION [8]. Let p be a fuzzy subgroup define a map p, : G + [0, l] by
on a group G. For any x E G,
b,(g)
= PW’)
VgEG
Then p, is called the fuzzy coset of p determined
by x.
The identity element of any group will always be denoted fuzzy subgroup of a group G, then it follows easily that P(X)
VXEG
by e. If p is a
(1)
3.21. Let p be a fuzzy subgroup of a group G. Let
x E G. Then
DEFINITION [8]. Let p be a fuzzy subgroup of a finite group G, and 9 be the set of all the distinct fuzzy cosets of ~1.Then the cardinality of 9 is called the index of CL. THEOREM 2.2 (Fuzzy Lagrange’s theorem [8, Theorem 4.101). Let p be Q fuzzy subgroup of a finite group G. Then the index of CLdivides the order of G. THFJOREM2.3 [8, Theorem 4.51. Given a fuzzy normal subgroup of a group G, let .F be the collection of all the fuzzy cosets of p. Then 9 (well-defined)
is a group under the
law of composition : P.Ay=Fxy
vx,y~G.
(4
FUZZY GROUPS: SOME?GROUP THEORE?TIC ANALOGS. II
79
Further, if ji : 9 -+ [0, l] if keened by
F(BJ =dO then ji is a fuzzy subgr~
VXEG,
on 9.
THEOREM 2.4 [8, Theorem 3.121. L.et p be a fuzzy normal subgroup of a group G with the identity element denoted, as usual, by e.Let H:={x~G:p(x)=p(e)}.
(3)
Then H is a normal subgroup of G. Define a map $: G/H --, [O,l] as follows (where G/H denotes, as usual, the quotient group of G module H):
Then /i is well defined and is a fuzzy normal subgroup of G/H. LEMMA2.5 [9, Lemma 3.75. Let p be a fuzzy subgroup of a group G. Define H:= {x~G:p(x)
=p(e)},
K:= {x~G:P~=jl,}, where e is the identity element of G. Then H = K and moreouer H is a s~gro~ of G. 3.
ANALOG OF CAYLEY’S THEOREM
-OREM 3.1. Let p be a fuzzy normal subgroup of a finite group G, and 3 be the set of fuzzy cosets of p. Then G has a natural permutation representation on the set .F. (Note that by Theorem 2.3, .F is a group with a binary operation given by 6%) tit H= {x~G:p(x)
-P(e)}.
Then H is a normal subgroup of G such that the quotient group G/H is isomorphic to the group .F. Proof. As remarked above, we have that 9 is a group. Suppose that G and 9 axe given by G= {gl,..., g,} and s={jig ,,..., ,i$}. For XEG, define a map
80
PRABIR BHA’ITACHARYA
AND N. P. MUKHERJEE
such that
(9 (i) rx is u permui~~~o~ of the set 9.
For, if
&=i+ then
a,,,(s) = P,,,(g)
WgeG
so
PL(W1d)=P(&gy-l) is atso an arbitrary
Since gx-’
p(ug,-‘) Therefore
proving
VgcG.
element of G, it follows then that =p(ug;‘)
V~EG.
we have
(i).
Now, let Sym(s) denote the symmetric group on 96- (that is, the group of all permutations of %). Let n : G + Sym( y) be the map defined by c(x) (ii) n is a homomorphism. g E G, we have that
= ?Tx
VXEG.
For, let x,y
be two elements
(6)
in G. Then for
FUZZY
GROUPS:
SOME GROUP
THEORETIC
ANALOGS.
II
81
since p is fuzzy normal. Again we have for g E G
(8) Comparing
(7) and (8), we get that
and thus Q is a homomorphism,
proving (ii).
(iii) The kernel of a is H. For, suppose that for some x E G we have that
where e denotes
the identity
of G. Then we have
a,) =a
= %@y) n
,.
Py* =
Py
Q~EG
byoiL = by
CB
QysG
Q~EG.
Since the set 9 = { 1;, : x E G} is a group, this gives us that
where $, is, as we know, the identity element of 9. Using Lemma 2.5, we have n now that x E H. Therefore the kernel of s is the subgroup H.
4.
FUZZY
ORDER
AND FUZZY
ABELIAN
First we describe some preliminary
results.
PROPOSITION4.1. Let p be a fuzzy subgroup of a group G. Then P is fuzzy normal - ~([x, y]) = p( e) Qx, y E G. (Notation: [x, y] = x-‘y-‘xy.)
PRABIR BHATTACHARYA AND N. P. MUKHERJEE
82
Proof. We have shown earlier [8, Theorem 3.41 that p is fuzzy normal * p is constant on the conjugate classes of G. Thus we have that p is fuzzy normal
(+ cs
p( x-“y-ix)
p(x-'Y-"xyY-') r(btvlv-‘1
= p( y-1)
tlx,y~G
=cl(r-')
Vx,yciG
=P(Y-l)
VX,_YEG
d[x,~l) =a(4
by Lemma 2.1.
COROLLARY 4.2. Lef ~1be a fuzzy n5r~ai subgroup of a group G. Let H=
=p(e)}.
{x~G:p(x)
Then H is a normaI subgroq of G and G’ c H, where G’ is the subgroup generated by ail elements [x, y] such that x, y E G. Further, the ~5tie~t group G/H is Abeiian. Proof. The fact that H is a normal subgroup of G follows from Theorem 2.4, Again Proposition 4.1 implies that G’ c H. Consequently G/H is Abelian. n
Now, we describe the motivation behind the two concepts of the order of a fuzzy subgroup and that of a fuzzy Abelian subgroup of a group G. Let K be a subgroup of a group 6. Then K is completely determined by the characteristic function XK:G-+ {O,l},
XK(X) =
0 1
if if
(9)
xQK, XE K.
Consider now the set H:=
{x~GIx~(x)
=xK(e)}.
Recall that for any subgroup X, the index of X in G, denoted by [G: X], is equal to the number of distinct oeft) cosets of X in G, which is also equal to
FUZZY
GROUPS:
SOME GROUP
THEORETIC
ANALOGS.
II
83
o(G)/o( X), where o(G) denotes the order ( - cardinality) of G, etc. Now, in the above situation it is clear that H - K, and the index of K in G is indeed equal to the index of H in G. Further, if [G : H] = n then we have that
o(H)=o(K)=--.
o(G)
(10)
Also, we note that H is Abelian if and only if K is Abelian. These considerations motivate us to consider the following analogs in the case of a fuzzy subgroup p of G. DEFINITION.Let p be a fuzzy subgroup of a finite group G. Then the order of p [notation o(p)] is defined by
o(G) o(P):=,--,
(11)
where r is the index of p (see Section 2 for the definition DEFINITION.Let p be a fuzzy subgroup H= (x~G:p(x) Then p is fuzzy Abelian if H is an Abelian
of index).
of a finite group G. Let =p(e)}. subgroup
(12) of G.
(Note that by Lemma 2.5, H is always a subgroup of G, but it need not be always Abelian.) From (11) it follows that if H is defined by (12), then
that is,
O(P)= 4H).
(13)
REMARK. If p is a fuzzy subgroup of a group G, then p is a map from G into the infinite set [O,l]. However, we are assigning a finite number as the “order” of the map p. The motivation behind this arrangement is clear if we compare (10) and (11).
84
PRABIR BHATTACHARYA The motivation
AND N. P. MUKHERJEE
behind the following defi~tion
stems from Zadeh 1121.
DEFINITION. Let IL, u be two fuzzy subgroups w
of the same group G. We say
If U, V are two subgroups of a finite group G such that o(U) = o(V) and U c_ V, then trivially U = V. However, if p, v are to fuzzy subgroups of a finite group G such that o(p) = o(v) and p & I, then it is always not necessary that p = v, as the following example shows: EXAMPLE 4.3. Let G be the “Klein four-group.” order 4 given by G = { e, a, b, ab}, where we have
and e is the identity that
of G. Choose three numbers
That is, G is a group of
t,, t,, t, belong to [O,l] such
t, > t, > t, . Define p : G --, [O,l] by setting
Cc(e) =t0,
da) = 4,
db) = (2,
It is easy to verify that p is a fuzzy subgroup numbers sO, sI, s2 belonging to [O,l] such that
p(ab)
= t,.
of G. Again,
choose three
$0 ’ St ’ $2 9 where to
t, < $3
(14)
t,
Now define v : G --, [O,l] by setting
v(e) =so,
v(a) =q.
v(b) =s*,
It is again easy to verify that v is a fuzzy subgroup
v(ab)
=s2.
of G. Further
it follows
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II
85
from (I4) that p 6 Y. Let H={xGG:p(x)-p(e))
and K=
(yEG:v(y)
xv(e)).
From the definitions of p and 8, it is clear that H = K. So by using (13) we get that
u(p) -o(H) =0(v). Hence we see that 1-1and Y have the same orders, p c z~,and yet clearly p + Y. RJ%ARK. The above example illustrates the fact that not all properties of a fuzzy subgroup could be expected to be analogs of results from group theory. The subgroup p defined in Example 4.3 is also an example of a fuzzy Abe&n subgroup, and the same is the case with the fuzzy subgroup P defined therein. We now give below an example of a fuzzy subgroup which is not fuzzy Abelian. EXAMPLE 4.4. Let G be a group of order 24 which is the direct product of the symmetric group S, and C,, the cyclic group of order 4. Obviously G is not Abelian. Now S, is a subgroup of G, and so is rt,, the dihedral group of order 12 (which is the group of symmetries of a regular hexagon). Further, we have that S, c De. Consider now the following chain of subgroups:
Choose three real numbers x19x2, x3 in [0, I] such that
Define a map A.: G -+ IO,11 by
It can be verified that A is a fuzzy subgroup of G (compare this with the main theorem in Bhattacharya [4], where the same type of construction is used to prove that given any chain of subgroups of a group G, there exists at least one fuzzy subgroup of G whose level subgroups are precisely the subgroups in this
PRABIR BHATTACHARYA
86
AND N. P. MUKHERIEE
chain). Now we have that
{xEG:A(x) =A(e)} is the subgroup Abelian.
S,. Hence it follows that A is not fuzzy Abelian,
since S, is not
REMARK. In [9] we gave a definition of fuzzy Abelian in the sense that p is fuzzy Abelian if ~([x, y]) = p(e) Vx, y E G, where [x, y] = x-‘y-ixy. Now Proposition 4.1 this definition is equivalent to just saying that Jo is fuzzy normal. Thus this definition is too specialized to be of much interest. Therefore we have recast the definition of a fuzzy Abelian subgroup here to give it a more broader meaning. We now prove an analog of a result from group theory that if U, Y are two subgroups of a group G such that U c V and V is Abelian, then U is Abelian. We have PROPOSITION 4.5. Let p, v be two fuzzy subgroups of a group G such that Y Q CL, v(e) = p(e), Proof.
and p is fuzzy Abelian. Then v is also fuzzy Abelian.
Let H,={xEG:p(x)=p(e)}
and H,={yEG:v(y)=v(e)}.
By hypothesis
Hi is Abehan.
Let y E Hz. Then we have that V(Y)
=v(e) =de).
By hypothesis V(Y) G P(Y).
so P(Y)a de). By (1) we have P(Y) 6
r(e).
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II
87
Therefore g(y) = pfe) and so y E Hi. Thus we obtain that Hz z HI and consequently H, is Abelian, implying that Y is fuzzy Abelian. 8 REMARK.If U, V are two subgroups of a group G, then e, the identity element of G, lies in both U and V. Consequently the characteristic functions xT1 and xr, [see (9)] assume identical values on e. Hence it is reasonable to assume in the hypothesis of Proposition 4.5 that p(e) = v(e). PRoposrnoN 4.6. A fuzzy subgroup of order p’, number, is fuzzy Abelian.
where p is some prime
Proof. Let p be a fuzzy subgroup of group G such that o(p) = p2. Let H={xEG:p(x)=p(e)}.
By (13) we have o(p) = u(H). Now it is a standard result in group theory that a group of order p2 is Abelian (in fact, Proposition 4.6 is an analog of this result which we are trying to establish here). Thus H is Abelian and consequently ,u is fuzzy Abelian. n For a finite group G, if H and K are two subgroups such that H C K, then it follows that the order of H divides the order of K- this result is, in fact, the classical Lagrange’s theorem. In Theorem 2.2 we have stated a fuzzy version of Lagrange’s theorem (which was proved in [8]). We now give another fuzzy version of this result, using the concept of order of a fuzzy subgroup. ?Z;IEOREM 4.7. Let /J and Y be two fuzzy subgroups of a finite group G such that p < v and p(e) = v(e). 7’hen the order of p divides the order of v.
Proof. Let H,=(xfG:p(x)=,u(e)}
and
As in the proof of Proposition 4.5, we obtain that HI c Hz. So by Lagrange’s theorem for finite groups we have that o( HI) divides o(H2). However, by (13) it follows that
4~) = o(H,),
o(v) =o(H,).
Consequently we get that the order of p divides the order of Y.
8
PRABIR BHATTACHARYA
88 5.
FUZZY
AND N. P. MUKHERJEE
SOLV~ILI~
The concept of a solvable group occupies a key role in group theory dating back to the last century. In this section we develop a fuzzy analog of solvability. First we need a preliminary result: LEMMA5.1. Let p, Y be twofuzzy normal subgroups of G such that v 6 p and v(e) = p(e). Then there exists a fuzzy normal subgroup q of a certain quotient group of G; q may be regarded as the fuzzy factor group p/v. Proof. Let K={x:v(x)=v(e)}, H-
(x:p(x)
=p(e)}.
Since ~1,Y are fuzzy normal, by Theorem 2.4 we have that both K and H are normal subgroups of G. Since v Q p, by arguing as in the proof of Proposition 4.5 we have that K E H. Now define a map 7:
G/K+
[O,l]
r)(xK) =4x>
(18)
VXEG.
We claim that q is well defined. For, suppose that xK = yK. Then xy-’ E K and so xy-’ E H, since K G H. Consequently we get that XH = yH.
(19)
Now by Theorem 2.4, p induces uniquely a map fi from G/H into [O,ll such that @(xH) = p(x). Therefore, using (19) we obtain that p(x) = a(y), which shows that the map 17is well defined. Further it is easy to see that n is a fuzzy n subgroup and f is fuzzy normal. DEFINITION.With the hypothesis of Lemma 5.1, the fuzzy subgroup defined by (18) is called the fuzzy ~tient group p,/v. Now consider
a chain of fuzzy normal subgroups
of G given by
9
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II where ~~(e)=~i(e),
l&i
89
Let
(21)
H,-(x~G:v~(~)==v~(e)}
Then we have that Hi is a normal subgroup of G since v, is fuzzy normal. It follows that we get a chain of normal subgroups of G corresponding to {20):
This series (22) will be used to characterize the series (20) of fuzzy normal subgroups. Thus for example let (22) be a series such that HI = G and H = { e}. This would of course imply that vi and v2 are given by
VI(X)=vl(e)
VXEG,
Vk(X)=“kw
only if
x=e.
By Lemma 5.1 it now follows that the series (20) yields another series of fuzzy normal subgroups: ha%>
*-* &‘k,
(23)
where ?i
=
‘i+l/“i
and q/q*i
= (xq+i
: XEG,
TiCx) -de)).
The above discussion motivates the followiug definition: DEFINITION.A fuzzy subgroup p of a group G is fmzy soluable if there exists a chain of fuzzy normal subgroups
with v*(x) = vk(e) only when x = e, and vi(e) = vi(e), 1 Q i < k, such that there is a corresponding chain of fuzzy normal subgroups
where qi := vivi+i and qi isf~yA~~~,
lgigk.
90
PRABIR BHA’ITACHARYA
AND N. P. MUKHERIEE
REMARKS.
(i) The chain of fuzzy normal subgroups given by (24) gives a chain normal subgroups {Hi} of the group G given by (22), where Hi is defined (21). Clearly Hk = {e}. (ii) If in the chain of fuzzy subgroups given by (24) we have that vi(x) vi(e) Vx E G, then from (21) we have that HI = G. So we have a chain subgroups H,=GaH,a.*.
of by = of
>H,={e},
where Hi/Hi + 1 is Abelian. Consequently G is solvable in this case. In the general situation when H1 is not necessarily equal to G, it is clear that H1 is always a solvable group. For a solvable group it is a standard result that any subgroup and any quotient group are both solvable. For a fuzzy solvable group we have the following analogs which are easy to establish, and we omit the details. -hIJiOREM
5.2.
(i) Let p be a fuzzy solvable subgroup of a group G, and v be a fuzzy subgroup of G such that p < v. Then v is also fuzzy solvable. (ii) If u and v are two fuzzy normal subgroups such that u is fuzzy solvable, then the quotient u/v is also fuzzy solvable. The first author is supported by Grant SM 042 from the Kuwait University, and the second author spent four weeks during M’arch- April 1986 as a Visiting Professor at the Kuwait University under the same grant, during which this paper was finalized.
REFERENCES 1. J. M. Anthony 69:124-130
and H. Sherwood, Fuzzy groups redefined,
J.
Math.
Anal.
Appl.
(1979).
2. J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and Systems 7:297-305 (1982). 3. P. Bhattacharya, Fuzzy subgroups: Some characterizations, J. Math. Anal. Appl., to
appear. 4. P. Bhattacharya, Fuzzy subgroups: Some characterizations II, Inform. Sci., to appear. 5. P. Bhattacharya and N. P. Mukhejee, Fuzzy groups and fuzzy relations, Inform. Sci. 36:267-282 (1985). 6. L. Biacino and G. Gerla, Closure systems and L-subalgebras, Inform. Sci. 33:181-195 (1984). 7. P. S. Das, Fuzzy groups and level subgroups, J. Math. Annl. Appl. 84:264-269 (1981).
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II 8. N. P. McKee 9. 10. 11. 12.
91
and P. 3~t~h~~ Fuzzy normal s~~~r~~ps and fuzzy cxxets, I~~fom?. Sd. 34:225-239 (1984). N. P. Mukhejee and P. Bbattacharya, Fuzzy groups: Some group theoretic analogs, &@wz. Sci., to appear. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35:X?-517 (1971). R. R. Yager (Ed.), Fuzzy Set and ~~sibj~i~ Tkeo~~ Recent ~v~~~~~e~f$, Pergm’~on~ 19x2. t. A. Zadeb, Fuzzy sets, Iform. und Control. 8:338-353 (1965).
Received I6 May 1986; revised 19 June
1986
SCIENCES
41, 77-91 (1987)
77
Fuzzy Groups Some Group ‘Ilteoretic Analogs. II PRABIR BHATTACHARYA
*
Depurtment of Muthemntics, Kuwait University, P.O. Box 5969, 13060 S&at, Kuwuit and N. P. MUKHERJEE School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi 110067, India Communicated
by Azriel Rosenfeld
ABSTRACT We introduce the concepts of (1) order of a fuzzy subgroup of a finite group, (2) a fuzzy Abelian group, and (3) a fuzzy solvable group. We obtain some results involving these notions which are analogs of results from group theory. Also, we prove analogs of two basic theorems of group theory, namely Cayley’s theorem and Lagrange’s theorem.
1.
INTRODUCTION
The present paper is a sequel to [9]. The concept of a fuzzy group was introduced by Rosenfeld in (lo]; he obtained a number of analogs of results from group theory. There have been some further ~vestigations to characterize fuzzy groups; see for example [l-2], [6], [7]. Through a series of papers [3-5, 8-91 we have attempted to investigate fuzzy groups and obtained several interesting characterizations. In particular we have obtained a number of fuzzy analogs of concepts from (finite) group theory and proved the fuzzy versions of some important group theoretical properties. Group theory plays a prominent role in mathematical sciences with a number of practical applications in areas like particle physics, coding theory and combinatorics, to name just a few. Our investigations have been motivated by interest in the error correcting capabilities of a linear code over a finite field. We obtain in the present paper the fuzzy analogs of two classical results from group theory, namely Cayley’s theorem and Lagrange’s theorem. We develop here the notions of the “order” of a fuzzy group and fuzzy Abelian group. Further we introduce the concept of a fuzzy solvable group. We obtain some analogs of group theoretical results related to these ideas introduced here. Our primary aim has been to indicate that fuzzy groups have a rich structure which *Present address: Department of Computer Science, University of Nebraska-Lincoln, Ferguson Bldg., Lincoln, NE 68588-0115. OElsevier Science Publishing Co., Inc. 1987 52 Vanderbilt Ave., New York, NY 10017
m20-0255/87/$03.50
PRABIR BHA’ITACHARYA
78
AND N. P. MUKHERJEE
is worth exploring, and some of the properties of a fuzzy group (but not all) are nice analogs of group theoretic results.
2.
PRELIMINARIES
We briefly review here some definitions Rosenfeld [lo] and our papers [3-51, [8-91.
and results. For more details see
DEFINITION [lo]. Let G be a group. A map p : G + [0, l] is called a fuzzy subgroup if (i) P(V) 2 ~NP(~),P(Y)) (ii) /.Qx-‘) = p(x) Vx E G.
vx, Y E G,
DEFINEION [8]. Let p be a fuzzy subgroup fuzzy normal if I = I Vx, y E G.
on a group G. Then ~1is called
DEFINITION [8]. Let p be a fuzzy subgroup define a map p, : G + [0, l] by
on a group G. For any x E G,
b,(g)
= PW’)
VgEG
Then p, is called the fuzzy coset of p determined
by x.
The identity element of any group will always be denoted fuzzy subgroup of a group G, then it follows easily that P(X)
VXEG
by e. If p is a
(1)
3.21. Let p be a fuzzy subgroup of a group G. Let
x E G. Then
DEFINITION [8]. Let p be a fuzzy subgroup of a finite group G, and 9 be the set of all the distinct fuzzy cosets of ~1.Then the cardinality of 9 is called the index of CL. THEOREM 2.2 (Fuzzy Lagrange’s theorem [8, Theorem 4.101). Let p be Q fuzzy subgroup of a finite group G. Then the index of CLdivides the order of G. THFJOREM2.3 [8, Theorem 4.51. Given a fuzzy normal subgroup of a group G, let .F be the collection of all the fuzzy cosets of p. Then 9 (well-defined)
is a group under the
law of composition : P.Ay=Fxy
vx,y~G.
(4
FUZZY GROUPS: SOME?GROUP THEORE?TIC ANALOGS. II
79
Further, if ji : 9 -+ [0, l] if keened by
F(BJ =dO then ji is a fuzzy subgr~
VXEG,
on 9.
THEOREM 2.4 [8, Theorem 3.121. L.et p be a fuzzy normal subgroup of a group G with the identity element denoted, as usual, by e.Let H:={x~G:p(x)=p(e)}.
(3)
Then H is a normal subgroup of G. Define a map $: G/H --, [O,l] as follows (where G/H denotes, as usual, the quotient group of G module H):
Then /i is well defined and is a fuzzy normal subgroup of G/H. LEMMA2.5 [9, Lemma 3.75. Let p be a fuzzy subgroup of a group G. Define H:= {x~G:p(x)
=p(e)},
K:= {x~G:P~=jl,}, where e is the identity element of G. Then H = K and moreouer H is a s~gro~ of G. 3.
ANALOG OF CAYLEY’S THEOREM
-OREM 3.1. Let p be a fuzzy normal subgroup of a finite group G, and 3 be the set of fuzzy cosets of p. Then G has a natural permutation representation on the set .F. (Note that by Theorem 2.3, .F is a group with a binary operation given by 6%) tit H= {x~G:p(x)
-P(e)}.
Then H is a normal subgroup of G such that the quotient group G/H is isomorphic to the group .F. Proof. As remarked above, we have that 9 is a group. Suppose that G and 9 axe given by G= {gl,..., g,} and s={jig ,,..., ,i$}. For XEG, define a map
80
PRABIR BHA’ITACHARYA
AND N. P. MUKHERJEE
such that
(9 (i) rx is u permui~~~o~ of the set 9.
For, if
&=i+ then
a,,,(s) = P,,,(g)
WgeG
so
PL(W1d)=P(&gy-l) is atso an arbitrary
Since gx-’
p(ug,-‘) Therefore
proving
VgcG.
element of G, it follows then that =p(ug;‘)
V~EG.
we have
(i).
Now, let Sym(s) denote the symmetric group on 96- (that is, the group of all permutations of %). Let n : G + Sym( y) be the map defined by c(x) (ii) n is a homomorphism. g E G, we have that
= ?Tx
VXEG.
For, let x,y
be two elements
(6)
in G. Then for
FUZZY
GROUPS:
SOME GROUP
THEORETIC
ANALOGS.
II
81
since p is fuzzy normal. Again we have for g E G
(8) Comparing
(7) and (8), we get that
and thus Q is a homomorphism,
proving (ii).
(iii) The kernel of a is H. For, suppose that for some x E G we have that
where e denotes
the identity
of G. Then we have
a,) =a
= %@y) n
,.
Py* =
Py
Q~EG
byoiL = by
CB
QysG
Q~EG.
Since the set 9 = { 1;, : x E G} is a group, this gives us that
where $, is, as we know, the identity element of 9. Using Lemma 2.5, we have n now that x E H. Therefore the kernel of s is the subgroup H.
4.
FUZZY
ORDER
AND FUZZY
ABELIAN
First we describe some preliminary
results.
PROPOSITION4.1. Let p be a fuzzy subgroup of a group G. Then P is fuzzy normal - ~([x, y]) = p( e) Qx, y E G. (Notation: [x, y] = x-‘y-‘xy.)
PRABIR BHATTACHARYA AND N. P. MUKHERJEE
82
Proof. We have shown earlier [8, Theorem 3.41 that p is fuzzy normal * p is constant on the conjugate classes of G. Thus we have that p is fuzzy normal
(+ cs
p( x-“y-ix)
p(x-'Y-"xyY-') r(btvlv-‘1
= p( y-1)
tlx,y~G
=cl(r-')
Vx,yciG
=P(Y-l)
VX,_YEG
d[x,~l) =a(4
by Lemma 2.1.
COROLLARY 4.2. Lef ~1be a fuzzy n5r~ai subgroup of a group G. Let H=
=p(e)}.
{x~G:p(x)
Then H is a normaI subgroq of G and G’ c H, where G’ is the subgroup generated by ail elements [x, y] such that x, y E G. Further, the ~5tie~t group G/H is Abeiian. Proof. The fact that H is a normal subgroup of G follows from Theorem 2.4, Again Proposition 4.1 implies that G’ c H. Consequently G/H is Abelian. n
Now, we describe the motivation behind the two concepts of the order of a fuzzy subgroup and that of a fuzzy Abelian subgroup of a group G. Let K be a subgroup of a group 6. Then K is completely determined by the characteristic function XK:G-+ {O,l},
XK(X) =
0 1
if if
(9)
xQK, XE K.
Consider now the set H:=
{x~GIx~(x)
=xK(e)}.
Recall that for any subgroup X, the index of X in G, denoted by [G: X], is equal to the number of distinct oeft) cosets of X in G, which is also equal to
FUZZY
GROUPS:
SOME GROUP
THEORETIC
ANALOGS.
II
83
o(G)/o( X), where o(G) denotes the order ( - cardinality) of G, etc. Now, in the above situation it is clear that H - K, and the index of K in G is indeed equal to the index of H in G. Further, if [G : H] = n then we have that
o(H)=o(K)=--.
o(G)
(10)
Also, we note that H is Abelian if and only if K is Abelian. These considerations motivate us to consider the following analogs in the case of a fuzzy subgroup p of G. DEFINITION.Let p be a fuzzy subgroup of a finite group G. Then the order of p [notation o(p)] is defined by
o(G) o(P):=,--,
(11)
where r is the index of p (see Section 2 for the definition DEFINITION.Let p be a fuzzy subgroup H= (x~G:p(x) Then p is fuzzy Abelian if H is an Abelian
of index).
of a finite group G. Let =p(e)}. subgroup
(12) of G.
(Note that by Lemma 2.5, H is always a subgroup of G, but it need not be always Abelian.) From (11) it follows that if H is defined by (12), then
that is,
O(P)= 4H).
(13)
REMARK. If p is a fuzzy subgroup of a group G, then p is a map from G into the infinite set [O,l]. However, we are assigning a finite number as the “order” of the map p. The motivation behind this arrangement is clear if we compare (10) and (11).
84
PRABIR BHATTACHARYA The motivation
AND N. P. MUKHERJEE
behind the following defi~tion
stems from Zadeh 1121.
DEFINITION. Let IL, u be two fuzzy subgroups w
of the same group G. We say
If U, V are two subgroups of a finite group G such that o(U) = o(V) and U c_ V, then trivially U = V. However, if p, v are to fuzzy subgroups of a finite group G such that o(p) = o(v) and p & I, then it is always not necessary that p = v, as the following example shows: EXAMPLE 4.3. Let G be the “Klein four-group.” order 4 given by G = { e, a, b, ab}, where we have
and e is the identity that
of G. Choose three numbers
That is, G is a group of
t,, t,, t, belong to [O,l] such
t, > t, > t, . Define p : G --, [O,l] by setting
Cc(e) =t0,
da) = 4,
db) = (2,
It is easy to verify that p is a fuzzy subgroup numbers sO, sI, s2 belonging to [O,l] such that
p(ab)
= t,.
of G. Again,
choose three
$0 ’ St ’ $2 9 where to
t, < $3
(14)
t,
Now define v : G --, [O,l] by setting
v(e) =so,
v(a) =q.
v(b) =s*,
It is again easy to verify that v is a fuzzy subgroup
v(ab)
=s2.
of G. Further
it follows
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II
85
from (I4) that p 6 Y. Let H={xGG:p(x)-p(e))
and K=
(yEG:v(y)
xv(e)).
From the definitions of p and 8, it is clear that H = K. So by using (13) we get that
u(p) -o(H) =0(v). Hence we see that 1-1and Y have the same orders, p c z~,and yet clearly p + Y. RJ%ARK. The above example illustrates the fact that not all properties of a fuzzy subgroup could be expected to be analogs of results from group theory. The subgroup p defined in Example 4.3 is also an example of a fuzzy Abe&n subgroup, and the same is the case with the fuzzy subgroup P defined therein. We now give below an example of a fuzzy subgroup which is not fuzzy Abelian. EXAMPLE 4.4. Let G be a group of order 24 which is the direct product of the symmetric group S, and C,, the cyclic group of order 4. Obviously G is not Abelian. Now S, is a subgroup of G, and so is rt,, the dihedral group of order 12 (which is the group of symmetries of a regular hexagon). Further, we have that S, c De. Consider now the following chain of subgroups:
Choose three real numbers x19x2, x3 in [0, I] such that
Define a map A.: G -+ IO,11 by
It can be verified that A is a fuzzy subgroup of G (compare this with the main theorem in Bhattacharya [4], where the same type of construction is used to prove that given any chain of subgroups of a group G, there exists at least one fuzzy subgroup of G whose level subgroups are precisely the subgroups in this
PRABIR BHATTACHARYA
86
AND N. P. MUKHERIEE
chain). Now we have that
{xEG:A(x) =A(e)} is the subgroup Abelian.
S,. Hence it follows that A is not fuzzy Abelian,
since S, is not
REMARK. In [9] we gave a definition of fuzzy Abelian in the sense that p is fuzzy Abelian if ~([x, y]) = p(e) Vx, y E G, where [x, y] = x-‘y-ixy. Now Proposition 4.1 this definition is equivalent to just saying that Jo is fuzzy normal. Thus this definition is too specialized to be of much interest. Therefore we have recast the definition of a fuzzy Abelian subgroup here to give it a more broader meaning. We now prove an analog of a result from group theory that if U, Y are two subgroups of a group G such that U c V and V is Abelian, then U is Abelian. We have PROPOSITION 4.5. Let p, v be two fuzzy subgroups of a group G such that Y Q CL, v(e) = p(e), Proof.
and p is fuzzy Abelian. Then v is also fuzzy Abelian.
Let H,={xEG:p(x)=p(e)}
and H,={yEG:v(y)=v(e)}.
By hypothesis
Hi is Abehan.
Let y E Hz. Then we have that V(Y)
=v(e) =de).
By hypothesis V(Y) G P(Y).
so P(Y)a de). By (1) we have P(Y) 6
r(e).
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II
87
Therefore g(y) = pfe) and so y E Hi. Thus we obtain that Hz z HI and consequently H, is Abelian, implying that Y is fuzzy Abelian. 8 REMARK.If U, V are two subgroups of a group G, then e, the identity element of G, lies in both U and V. Consequently the characteristic functions xT1 and xr, [see (9)] assume identical values on e. Hence it is reasonable to assume in the hypothesis of Proposition 4.5 that p(e) = v(e). PRoposrnoN 4.6. A fuzzy subgroup of order p’, number, is fuzzy Abelian.
where p is some prime
Proof. Let p be a fuzzy subgroup of group G such that o(p) = p2. Let H={xEG:p(x)=p(e)}.
By (13) we have o(p) = u(H). Now it is a standard result in group theory that a group of order p2 is Abelian (in fact, Proposition 4.6 is an analog of this result which we are trying to establish here). Thus H is Abelian and consequently ,u is fuzzy Abelian. n For a finite group G, if H and K are two subgroups such that H C K, then it follows that the order of H divides the order of K- this result is, in fact, the classical Lagrange’s theorem. In Theorem 2.2 we have stated a fuzzy version of Lagrange’s theorem (which was proved in [8]). We now give another fuzzy version of this result, using the concept of order of a fuzzy subgroup. ?Z;IEOREM 4.7. Let /J and Y be two fuzzy subgroups of a finite group G such that p < v and p(e) = v(e). 7’hen the order of p divides the order of v.
Proof. Let H,=(xfG:p(x)=,u(e)}
and
As in the proof of Proposition 4.5, we obtain that HI c Hz. So by Lagrange’s theorem for finite groups we have that o( HI) divides o(H2). However, by (13) it follows that
4~) = o(H,),
o(v) =o(H,).
Consequently we get that the order of p divides the order of Y.
8
PRABIR BHATTACHARYA
88 5.
FUZZY
AND N. P. MUKHERJEE
SOLV~ILI~
The concept of a solvable group occupies a key role in group theory dating back to the last century. In this section we develop a fuzzy analog of solvability. First we need a preliminary result: LEMMA5.1. Let p, Y be twofuzzy normal subgroups of G such that v 6 p and v(e) = p(e). Then there exists a fuzzy normal subgroup q of a certain quotient group of G; q may be regarded as the fuzzy factor group p/v. Proof. Let K={x:v(x)=v(e)}, H-
(x:p(x)
=p(e)}.
Since ~1,Y are fuzzy normal, by Theorem 2.4 we have that both K and H are normal subgroups of G. Since v Q p, by arguing as in the proof of Proposition 4.5 we have that K E H. Now define a map 7:
G/K+
[O,l]
r)(xK) =4x>
(18)
VXEG.
We claim that q is well defined. For, suppose that xK = yK. Then xy-’ E K and so xy-’ E H, since K G H. Consequently we get that XH = yH.
(19)
Now by Theorem 2.4, p induces uniquely a map fi from G/H into [O,ll such that @(xH) = p(x). Therefore, using (19) we obtain that p(x) = a(y), which shows that the map 17is well defined. Further it is easy to see that n is a fuzzy n subgroup and f is fuzzy normal. DEFINITION.With the hypothesis of Lemma 5.1, the fuzzy subgroup defined by (18) is called the fuzzy ~tient group p,/v. Now consider
a chain of fuzzy normal subgroups
of G given by
9
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II where ~~(e)=~i(e),
l&i
89
Let
(21)
H,-(x~G:v~(~)==v~(e)}
Then we have that Hi is a normal subgroup of G since v, is fuzzy normal. It follows that we get a chain of normal subgroups of G corresponding to {20):
This series (22) will be used to characterize the series (20) of fuzzy normal subgroups. Thus for example let (22) be a series such that HI = G and H = { e}. This would of course imply that vi and v2 are given by
VI(X)=vl(e)
VXEG,
Vk(X)=“kw
only if
x=e.
By Lemma 5.1 it now follows that the series (20) yields another series of fuzzy normal subgroups: ha%>
*-* &‘k,
(23)
where ?i
=
‘i+l/“i
and q/q*i
= (xq+i
: XEG,
TiCx) -de)).
The above discussion motivates the followiug definition: DEFINITION.A fuzzy subgroup p of a group G is fmzy soluable if there exists a chain of fuzzy normal subgroups
with v*(x) = vk(e) only when x = e, and vi(e) = vi(e), 1 Q i < k, such that there is a corresponding chain of fuzzy normal subgroups
where qi := vivi+i and qi isf~yA~~~,
lgigk.
90
PRABIR BHA’ITACHARYA
AND N. P. MUKHERIEE
REMARKS.
(i) The chain of fuzzy normal subgroups given by (24) gives a chain normal subgroups {Hi} of the group G given by (22), where Hi is defined (21). Clearly Hk = {e}. (ii) If in the chain of fuzzy subgroups given by (24) we have that vi(x) vi(e) Vx E G, then from (21) we have that HI = G. So we have a chain subgroups H,=GaH,a.*.
of by = of
>H,={e},
where Hi/Hi + 1 is Abelian. Consequently G is solvable in this case. In the general situation when H1 is not necessarily equal to G, it is clear that H1 is always a solvable group. For a solvable group it is a standard result that any subgroup and any quotient group are both solvable. For a fuzzy solvable group we have the following analogs which are easy to establish, and we omit the details. -hIJiOREM
5.2.
(i) Let p be a fuzzy solvable subgroup of a group G, and v be a fuzzy subgroup of G such that p < v. Then v is also fuzzy solvable. (ii) If u and v are two fuzzy normal subgroups such that u is fuzzy solvable, then the quotient u/v is also fuzzy solvable. The first author is supported by Grant SM 042 from the Kuwait University, and the second author spent four weeks during M’arch- April 1986 as a Visiting Professor at the Kuwait University under the same grant, during which this paper was finalized.
REFERENCES 1. J. M. Anthony 69:124-130
and H. Sherwood, Fuzzy groups redefined,
J.
Math.
Anal.
Appl.
(1979).
2. J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and Systems 7:297-305 (1982). 3. P. Bhattacharya, Fuzzy subgroups: Some characterizations, J. Math. Anal. Appl., to
appear. 4. P. Bhattacharya, Fuzzy subgroups: Some characterizations II, Inform. Sci., to appear. 5. P. Bhattacharya and N. P. Mukhejee, Fuzzy groups and fuzzy relations, Inform. Sci. 36:267-282 (1985). 6. L. Biacino and G. Gerla, Closure systems and L-subalgebras, Inform. Sci. 33:181-195 (1984). 7. P. S. Das, Fuzzy groups and level subgroups, J. Math. Annl. Appl. 84:264-269 (1981).
FUZZY GROUPS: SOME GROUP THEORETIC ANALOGS. II 8. N. P. McKee 9. 10. 11. 12.
91
and P. 3~t~h~~ Fuzzy normal s~~~r~~ps and fuzzy cxxets, I~~fom?. Sd. 34:225-239 (1984). N. P. Mukhejee and P. Bbattacharya, Fuzzy groups: Some group theoretic analogs, &@wz. Sci., to appear. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35:X?-517 (1971). R. R. Yager (Ed.), Fuzzy Set and ~~sibj~i~ Tkeo~~ Recent ~v~~~~~e~f$, Pergm’~on~ 19x2. t. A. Zadeb, Fuzzy sets, Iform. und Control. 8:338-353 (1965).
Received I6 May 1986; revised 19 June
1986