In this paper, we have established some independent proof of homomorphism on algebra of abelian and characteristic fuzzy subgroup. The characteristic of fuzzy subgroup  was first introduced by P. Bhattacharya and N. P. Mukharjee in 1986.
The concept of fuzzy sets was introduced by L.A.Zadeh  in 1965.Study of algebraic structure was first introduced by A.Rosenfeld . After that a series of researches have done in this direction P.Bhattacharya and N.P.Mukharjee have defined fuzzy normal subgroup and characteristic fuzzy subgroup in 1986. In this paper we have tried to established some independent proof about the properties of fuzzy group homomorphism on algebra of characteristic fuzzy subgroup.
In this section, we recall and study some concepts associated with fuzzy sets and fuzzy group, which we need in the subsequent sections.
A. Fuzzy Set
Over the past three decades, a number of definitions of a fuzzy set and fuzzy group have appeared in the literature (cf., e.g., [15, 1, 3, 7, 10]). In , it has been shown that some of these are equivalent. We begin with the following basic concepts of fuzzy set, fuzzy point and fuzzy group.
 A.Rosenfeld.: Fuzzy group. J.Math.Anal.Appl.35,512-517 (1971).
 S.Sebastian and S.Babunder. Fuzzy groups and group homomorphism.Fuzzy sets and systems.8,397-401 (1996).
 S. Abou-zaid, On fuzzy subgroup, Fuzzy sets and systems,55. 1993, pp. 237–240.
 N.Ajmal and A.S.Prajapati,Fuzzy cosets and fuzzy normal subgroup, Inform.Sci, 64 (1992) 17-25.
 R.Biswas,Fuzzy subgroup and anti fuzzy subgroup.Fuzzy sets and systems.35,121-124 (1990).
 D.S.Malik,and J.N.Mordeson,Fuzzy subgroup and abelian group.Chinese.J.Math(Taipei).19,129-145 (1991).
 J.M.Anthony,and H.Sherwood,Fuzzy subgroup redefined,J.Math.Anal.Appl.69,124-130 (1979).
 J.M.Anthony,and H.Sherwood,A characterization of fuzzy subgroup, J.Math.Anal.Appl,69 (1979) 297-305.
 P.S.Das,Fuzzy groups and level subgroups J.Math.Anal.Appl.,84 (1981) 264-269.
 V.N.Dixit,R.kumar,and N.Ajmal,Level subgroups and union of fuzzy subgroups. Fuzzy Sets and Systems, 37 (1990) 359-371. and Technology, 1994, Vol. 2, pp. 87-98.
 M.S.Eroglu, The homomorphic image of a fuzzy subgroup is always a fuzzy subgroup, Fuzzy Sets and Systems, 33 (1989) 255-256.
 I.J.Kumar,P.K.Saxena,and P.Yadav,Fuzzy normal subgroups and fuzzy quotients, Fuzzy sets and systems, 46 (1992) 121-132.
 N.P.Mukherjee,and P.Bhattacharya,Fuzzy normal subgroups and fuzzy cosets: Information Science, 34 (1984) 225-239.
 M.T.A.Osman, On the direct product of fuzzy subgroups, Fuzzy sets and sysems, 12 (1984) 87-91.
 L.A.Zadeh. Fuzzy sets, Inform.Control, 8 (1965) 338-353.
 S.M.A.Zaidi, and Q.A.Ansari,: Some results of categories of L-fuzzy subgroups.Fuzzy sets and systems, 64 (1994) 249-256. Information Technology, 2012, Vol. 2, pp. 527-531. automata. Soft Computing, 2013 (Communicated).
 Y.Yu, A theory of isomorphism of fuzzy groups, Fuzzy system and Math.2,(1988),57-68.