FUZZIFICATION OF CAYLEY'S AND LAGRANGE'S THEOREMS

8/10/2019 FUZZIFICATION OF CAYLEY'S AND LAGRANGE'S THEOREMS

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FUZZIFICATION OF CAYLEY'S AND LAGRANGE'STHEOREMS

Thampy Abraham* and Souriar Sebastian**

*Department of MathematicsSt. Peter's College, Kolenchery -682 311, Kerala (India)

E-mail: [email protected]

**Department of MathematicsSt. Albert's College, Ernakulam-682 018, Kerala (India)

ABSTRACT

The notion of fuzzy subsets was introduced by L.A.Zadehin 1965.In 1971, Rosenfeld defined the fuzzy subgroups and gavesome of its properties. Fuzzy versions of various algebraic structures

have been studied by Mathematicians like Abu Osman, Katsarasand Liu and Gu Wenxiang. G. Frobenius developed the theory of grouprepresentations at the end of the 19th century. As a continuation of these works, in this paper, we fuzzify the famous theorems due toCayley and Lagrange in group theory in a different way. The theoryof group representations has applications in several branches of Mathematics and practical Physics. So the study of its fuzzy versionis expected to have many practical applications.

Key words: Fuzzy subroup, Fuzzy homomorphism, Fuzzyrepresentation, Fuzzy order.

J. Comp. & Math. Sci. Vol. 1(1), 41-46 (2009).

1. INTRODUCTION

The notion of fuzzy subsets wasintroduced by L.A. Zadeh14 in 1965. In 1971,Rosenfeld 7 defined the fuzzy subgroups and gave some of its properties. Fuzzy versions of various algebraic structures have been studied by Mathematicians like Abu Osman, Katsarasand Liu and Gu Wenxiang. Reports of someworks done by the author/s in this area are

contained in 9 to 13. G. Frobenius developed the theory of group representations at the end of the 19th century. As a continuation of theseworks, in this paper, we fuzzify the famoustheorems due to Cayley and Lagrange in grouptheory in a different way. The theory of grouprepresentations has applications in several branches of Mathematics and practical Physics.So the study of its fuzzy version is expected tohave many practical applications.


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2. Preliminaries :

Let X be a non-empty set. A function : X [0,1] is called a fuzzy subset of X.The set of all fuzzy subsets of X is usuallydenoted by F(X ).

2.1. Definition 7. A fuzzy subset of a group G is called a fuzzy subgroup of G if,for all x, y G, (xy) (x) (y) and (x -1)= (x).

2.2. Definition 2,9. Let be a fuzzysubset of a set S. Then for t [0,1],the set t = {x S | (x) t} is called a level subsetof .

The level subsets t, t [0, (e)] of a

fuzzy subset are called level subgroups of in G. It can be seen that the level subgroups

of a given fuzzy subgroup form a chain.

2.3 Definition 6. A fuzzy subgroupof G is called a fuzzy normal subgroup if

(xy)= (yx) for all x,y G.

2.4 Definition 3. Let f be a functiondefined from X to Y. The image of a fuzzysubset on X under f is the fuzzy subset f ( )of Y defined by

f ()(y) = V{(x): x f -1(y)} for all y R(f)= 0, otherwise

The pre-image of the fuzzy subseton Y under f is fuzzy subset f -1 ( ) of Xdefined by

f -1 ( )(x) = V{ f (x)}, x X

2.5 Definition 4. Let be a fuzzy

subgroup of a group G. Given x G, the least positive integer n such that (xn) = (e) iscalled the fuzzy order of x with respect to .If no such n exists, x is said to have infinitefuzzy order with respect to . The fuzzy order of x with respect to is denoted by FO (x).

2.6 Defini tion 4. Let be a fuzzysubgroup of G. The least positive integer n suchthat (xn) = (e) for all x G, is called the

order of , denoted by O( ). If no such n exists, is said to have an infinite order.

2.7. Proposition 4. Let be a fuzzysubgroup of a group G. If (xm) = (e) for some x G, then FO (x) divides m.

2.8. Proposition. Let be a fuzzysubgroup of G. If (xm) = (e) for all x G,then FO (x)|m. Proof follows from 2.7.

2.9. Theorem 6. Let f be a homomorphism defined from a group G into a group G'.Then for every fuzzy subgroup of G, f ( ) isa fuzzy subgroup of G'.

2.10. Theorem 6. Let f be a homomorphism defined from a group G into a groupG'.Then for every fuzzy subgroup of G', f -1

( ) is a fuzzy subgroup of G.

2.11. Definition 1. Let G be a groupand M be a vector space over a field K . Alinear representation of G with representationspace M is a homomorphism of G into GL(M)where GL(M) is a group of units in Homk (M,M), called the general linear group.

2.12. Definition 6. Let G and G' begroups. Let be a fuzzy group on G and be


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subgroup of G and let N be a normal subgroupof G. Define by {[x]} = v { (z) | z [x]}for all x G, where [x] denotes the coset xN.Then is a fuzzy subgroup of G/N.

The fuzzy subgroup defined on G/N

is called the quotient fuzzy subgroup or factor fuzzy subgroup of the fuzzy subgroup of Grelative to the normal subgroup N of G and isdenoted by /N.

2.17. A fundamental theorem of fuzzy representations 10. Let G be a group and M be a vector space over a filed K . If T : GGL(M) is a fuzzy representation of G, then

: G/N GL(M) is a fuzzy representation of G/N where N is a normal subgroup of G.

2.18. Corollory 10. Let T be a homo-morphism of a group G into a group G' and T be a fuzzy homomorphism of onto where

is a fuzzy subgroup of G and is a fuzzysubgroup of T(G). Then : G/N G' is afuzzy homomorphism of onto where is afuzzy subgroup of G/N and N is a normalsubgroup of G.

2.19. Corollory 10. Let T be a homo-morphism of a group G onto a group G' and K

be the kernel of T . If T is a fuzzy homomorphism of onto where is a fuzzy subgroupof G and is a fuzzy subgroup of G', then :G/K G' is a fuzzy isomorphism of ontowhere is a fuzzy subgroup of G/K.

3. Fuzzification of Cayley's and Lagrange'sTheorems :

We know that T is a fuzzy represen-tation of a group G with representation spaceM if T is a homomorphism of G into the general

linear space GL(M) and T( ) = , where isa fuzzy subgroup of G and n is a fuzzy subgroupof T(G).When GL(M) is replaced by a groupG', T will be merely a fuzzy homomorphism of

onto . In this section we analyse the fuzzyhomomorphism between fuzzy groups and

order of a fuzzy group and try to fuzzify thefamous theorems due to Cayley and Lagrangein group theory.

3.1. Cayley's theorem 5. Every finitegroup is isomorphic to a subgroup of A(s) for some appropriate set S, where A(S) is the setof all automorphisms of S.

3.2. Fuzzification of Cayley's theorem .If is an isomorphism of a finite group G ontoa subgroup of A(G),the group of automor-

phisms of G, then is a fuzzy isomorphism of onto where is a fuzzy subgroup of G and

n is a fuzzy subgroup of (G).

Proof:

Let be a fuzzy subgroup of G and n be a fuzzy subgroup of (G) defined by (tg)= (g), for g G, tg A(G). Let be theisomorphism defined from G to (G) by (g)= tg, where tg is an automorphism

( )(tg) = V{ (g) | g -1(tg)} = (g), since s an isomorphismFor tg , th (G), ( ) (tg th)= ( )tgh = (gh)

(g) (h) ( ) (tg) ( ) (th)

( ) (tg)-1 = ( ) (tg-1) = (g-1) = (g) = ( )(tg).


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Therefore is a fuzzy isomorphism of onto.

3.3. Example. Let G = {1,-1}, (G)= A(G) = {t1,t -1},

: G A (G) be defined by (g) = tg . Then is an isomorphism.

Consider on G by (1) = 1, (-1) =1/2. Then is a fuzzy subgroup of G. Let bea fizzy subgroup defined on the range A(G) of

by (t1) = 1 and (t-1)=1/2. We have to showthat is a fuzzy isomorphism. i.e to provethat ( ) = .

( ) (t1) = V{ (x) | x -1(t1)} = 1 ( ) (t-1)= V{ (x) | x -1(t-1)} = 1/2

Hence ( )= . Therefore is a fuzzy isomorphism of onto .

3.4. Lagrange's Theorem 5. If G is afinite group and H is a subgroup of G, thenO(H) |O(G) .

3.5. Theorem. Let H be a subgroupof a group G and let n be the order of a fuzzysubgroup of G. If O (|H) exists then O (|H)| O().

Proof : Let O( ) = n. Then (xn) =(e) x G.

H is a subgroup of G. Therefore for x H, x, x2, .. H

If O (|H) exists , then O (|H) n. If O (|H) = n, then O (|H) | O ( ).

O (|H) < n, let O (|H) = m.

Then (|H) (xm) = (e) x H, if(xm) = (e) x H. Then m is the fuzzy order of at least one element x in H. Now FO (x)=m and (xm)= (e) x G. So by proposition2.7, m I n i.e. , O (|H) | O ( ).

3.6. Definition 1. Two representationsT and T' with spaces M and M ' are said to beequivalent if there exists a K -isomorphism S of M onto M ' such that T'(g)(S)=ST(g), g G

3.7. Definition. Let G be a group and be a fuzzy subgroup of G. Two fuzzy represen-tations T and T' of G with spaces M and M 'are said to be equivalent if

T -1 ( )(x) = T '-1( )(x),

where and are fuzzy subgroups defined on T (G) and T' (G) respectively.

3.8. Remark. The relation 'equivalent'of fuzzy representations is an equivalencerelation.

3.9. Defini tion. Let T be a fuzzyrepresentation of a group G with representationspace M and N be a subgroup of G. Then therestriction of T on N is defined as

(T | N)(x) = T (x) , x N

3.10. Proposition . If T is a fuzzyrepresentation of G with representation spaceM and N is a subgroup of G, T| N is a fuzzyrepresentation on N.

Proof: Since T is a fuzzy represen-tation, there exists fuzzy subgroups and of G and T(G) such that T( ) = . We have to


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show that T | N is a fuzzy representation.

For y T (G),

(T | N) ( )(y) = V { (x) | x (T | N)-1 (y)} = V { (x) | (T | N) (x) = y}

= V{ (x) | T (x) = y, x N} = V{ (x) | T (x)=y, y T(N)} = | T (N) (y). ( T | N) ( ) = | T (N) .

Therefore T| N is a fuzzy homomorphism of | Nonto |T (N). .Hence T| N is a fuzzy representationof | N onto |T (N).

REFERENCES

1. Charles W. Curties and Irving Reiner, Representation Theory of Finite groupand Associative Algebras, INC, (1962).

2. P.S. Das, Fuzzy groups and level groups, J. Math. Anal. Appli. 84, 264-269 (1981).

3. George J. Klir and Bo Yuan, Fuzzy setsand Fuzzy logic, Prentice-Hall of India(2000).

4. Jae-Gyeom Kim and Hau Duo Kim, Infor-mation Sciences, 80, 243-252 (1994).

5. John B. Fraleigh, A first course in AbstractAlgebra, Third Edition, Addison-Wesley,(1999).

6. John N. Mordeson and D.S. Malik,FuzzyCommunicative Algebra, World ScientificPublishing (1998).

7. A. Rosenfield, Fuzzy groups, J. M. Anal. Appli. 35, 512-517 (1971).

8. Sherry Fernandez, Fuzzy G-modules and

Fuzzy representations, TAJOPAM 1, 107-114 (2002).9. Souriar Sebastian and S. Babu Sundar, On

the chains of level subgroups of homomorphicimages and pre-images of Fuzzy subgroups,

Banyan Mathematical Journal 1 , 25-34(1994).

10. Souriar Sebastain, Mercy K. Jacob and Thampy Abraham, A fundamental theoremof fuzzy representation, TAJOPAM (accepted).

11. Souriar Sebastian and S. Babu Sundar,

Existence of fuzzy sub-groups of every levelcardinality upto Ho, Fuzzy sets and Systems,67, 365-368 (1994).

12. Souriar Sebastian and S. Babu Sundar,Generalisations of some results of Das,Fuzzy sets and Systems, 71 , 251-253(1995).

13. Souriar Sebastian and S. Babu Sundar,Fuzzy groups and group homomorphisms,Fuzzy sets and Systems, 81, 397-401(1996).

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FUZZIFICATION OF CAYLEY'S AND LAGRANGE'S THEOREMS