Higher Indicators for the Doubles of some Totally Orthogonal Groups

TL;DR

This paper studies the indicators of certain group doubles, revealing an infinite family of totally orthogonal groups with modules exhibiting negative second indicators, thus addressing key questions in the theory of group doubles.

Contribution

It introduces an infinite family of totally orthogonal, completely real groups generated by involutions, with doubles that admit modules with second indicator -1.

Findings

Identified groups with second indicator -1 in their doubles

Constructed an infinite family of totally orthogonal groups

Provided answers to questions about doubles of totally orthogonal groups

Abstract

We investigate the indicators for certain groups of the form $\BZ_k\rtimes D_l$ and their doubles, where $D_l$ is the dihedral group of order $2l$. We subsequently obtain an infinite family of totally orthogonal, completely real groups which are generated by involutions, and whose doubles admit modules with second indicator of -1. This provides us with answers to several questions concerning the doubles of totally orthogonal finite groups.