Some properties of group-theoretical categories

TL;DR

This paper explores properties of group-theoretical categories, including their grading, nilpotency conditions, simple objects, invertible objects, and universal grading groups, providing explicit descriptions and criteria within the framework of group theory.

Contribution

It introduces a grading by a double coset ring, characterizes nilpotency, and describes simple and invertible objects explicitly in group-theoretical terms.

Findings

Every group-theoretical category is graded by a double coset ring.

Provides a necessary and sufficient condition for nilpotency.

Explicit descriptions of simple objects and invertible objects in group-theoretical terms.

Abstract

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.