Strong Gelfand Pairs of the Dihedral and Dicyclic Groups

TL;DR

This paper classifies all strong Gelfand pairs involving dihedral and dicyclic groups, identifying subgroup pairs where irreducible characters induce multiplicity-free representations.

Contribution

It provides a complete characterization of strong Gelfand pairs for dihedral and dicyclic groups, expanding understanding of their representation theory.

Findings

Identifies all strong Gelfand pairs in dihedral groups.

Identifies all strong Gelfand pairs in dicyclic groups.

Provides explicit subgroup classifications for these pairs.

Abstract

A strong Gelfand pair $(G, H)$ is a finite group $G$ and a subgroup $H$ where every irreducible character of $H$ induces to a multiplicity-free character of $G$. We determine the strong Gelfand pairs of the dihedral groups $D_{2n}$ and the dicyclic groups $Dic_{4n}$ for all $n$.