Abstract The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. generalised says that the probability an element occurring in image a map group G is either 0, or at least 1/| |. Noting work Ashurst, we name Amit–Ashurst and show holds p -groups with cyclic maximal subgroup.

fuzzy groups and fuzzy theory have a lot of applications in several sciencessuch as mathematics, computer science, computer and electrical engineering.hence, counting the number of fuzzy subgroups of finite groups to classify them isan important issue in fuzzy theory. the main goal of this paper is to give an explicitformula for the number of fuzzy subgroups of a finite cyclic groupg z p1 z p 2...