  We establish some relations between the orders of simple objects in a fusion
category and the structure of its universal grading group. We consider fusion
categories which have a faithful simple object and show that its universal
grading group must be cyclic. As for the converse, we prove that a braided
nilpotent fusion category with cyclic universal grading group always has a
faithful simple object. We study the universal grading of fusion categories
with generalized Tambara-Yamagami fusion rules. As an application, we classify
modular categories in this class and describe the modularizations of braided
Tambara-Yamagami fusion categories.
