Primary fuzzy subgroups

The concept of the fuzzy order of an element of a group relative to a fuzzy subgroup was introduced by Suryansu Ray {[}7]. Here we prove that every element of a group and its inverse have the same fuzzy order and that the set of all elements of a finite fuzzy order is a subgroup and defines the p-component of a group relative to the fuzzy subgroup. Also, we define the order of a fuzzy subgroup and prove the Lagrange's Theorem in the fuzzy case and give a counterexample for its famous corollary, define the primary fuzzy subgroup and prove some of their properties, some of which are analogous with the crisp case, and explains some of these properties which are invalid in the fuzzy case. (C) 2000 Elsevier Science B.V. All rights reserved.