The Remak-Krull-Schmidt Theorem on\\ Fuzzy Groups

Document Type : Research Paper


1 Department of Mathematics, University of Fort Hare, Alice 5700 , Eastern Cape , South Africa

2 Department of Mathematics ( Pure & Applied ), Rhodes University, Grahamstown 6140, Eastern Cape, South Africa


In this paper we study a representation of a fuzzy subgroup $\mu$ of a group $G$, as a product of indecomposable fuzzy subgroups called the components of $\mu$.  This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and some of their properties to prove this theorem. We illustrate with some examples.


P. M. Cohn, {it Universal algebra}, D. Reidal Publishing Com., Boston, 1965.

P. S. Das, {it Fuzzy groups and level subgroups}, J. Math. Anal. and Appl., {bf 84}textbf{(1)} (1981), 264--269.

A. Jain, {it Fuzzy subgroups and certain equivalence relations}, Iranian Journal of Fuzzy Systems, {bf 3}textbf{(2)} (2006), 75--91.

A. G. Kurosh, {it Theory of groups}, Chelsea Pub., New York, 1956.

B. B. Makamba, {it Studies in fuzzy groups}, Doctoral Thesis, Rhodes University, South Africa, 1992.

 N. P. Mukherjee and P. Bhattacharya, {it Fuzzy normal subgroups and fuzzy cosets}, Information Sciences, {bf 34}textbf{(3)} (1984), 225--239.

V. Murali and B. B. Makamba, {it On an equivalence of fuzzy subgroups I}, Fuzzy Sets and Systems, {bf 123}textbf{(2)} (2001), 259--264.

O. Ore,  {it On the foundation of abstract algebra II}, Ann. of math.(2), {bf 37}textbf{(2)} (1936), 265--292.

R. Remak, {it Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren ("On the decomposition of a finite group into indirect indecomposable factors")}, Thesis, Humboldt University of Berlin, 1911.

A. Rosenfeld, {it Fuzzy groups}, J. Math. Anal. Appl., {bf 35} (1971), 512--517.

H. Sherwood, {it Product of fuzzy groups}, Fuzzy Sets and Systems, {bf 11}textbf{(1)} (1983), 79--89.

Y. Zhang  and K. Zou, {it A note on an equivalence relation on fuzzy subgroups}, Fuzzy Sets and Systems, {bf 95}textbf{(2)} (1998), 243--247.