Some properties of group-theoretical categories

Understand

We first show that every group-theoretical category is graded by a certain double coset ring.

• As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent.
• We then give an explicit description of the simple objects in a group-theoretical category (following Ostrik, arXiv:math/0202130) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms.
• Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category.