Transposable character tables, dual groups

TL;DR

This paper explores a generalized notion of duality for finite groups via their character tables, extending the self-duality concept from Abelian groups to certain noncommutative groups, and examines the structural implications of such dualities.

Contribution

It introduces a new concept of duality for noncommutative groups based on character table transposition proximity and analyzes the structural properties of dual groups.

Findings

Dual groups have dual normal subgroup lattices.

The duality concept applies to some p-groups.

Cannot be extended to non-nilpotent groups.

Abstract

One way of expressing the self-duality $A\cong \Hom(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). Noncommutative groups fail to satisfy this property. In this paper we extend the duality to some noncommutative groups considering when the character table of a finite group is close to being the transpose of the character table for some other group. We find that groups dual to each other have dual normal subgroup lattices. We show that our concept of duality cannot work for non-nilpotent groups and we describe $p$-group examples.