Higher indicators for some groups and their doubles

TL;DR

This paper explicitly calculates the indicators for certain non-abelian groups, including dihedral and quaternion groups, and their Drinfel'd doubles, revealing that indicators are integers with specific conditions for negatives.

Contribution

It provides explicit formulas for indicators of groups formed as semidirect products of cyclic groups and quaternion groups, expanding understanding of their algebraic properties.

Findings

Indicators are all integers for these groups.

Negative indicators occur only under specific conditions.

The results include dihedral, semidihedral, and quaternion groups.

Abstract

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups includes the dihedral groups, the non-abelian groups of order $pq$, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.