On involutions and generalized symmetric spaces of dicyclic groups

TL;DR

This paper characterizes automorphisms of dicyclic groups, especially involutions, and describes their associated generalized symmetric spaces, fixed point groups, and orbit structures under twisted conjugation.

Contribution

It provides a detailed description of automorphisms of dicyclic groups and analyzes the structure of their generalized symmetric spaces and fixed point groups, focusing on involutions.

Findings

Explicit description of automorphisms of dicyclic groups

Characterization of fixed point groups for involutions

Analysis of orbit structures under twisted conjugation

Abstract

Let $G=\Dc_{n}$ be the dicyclic group of order $4n$. Let $\varphi$ be an automorphism of $G$ of order $k$. We describe $\varphi$ and the generalized symmetric space $Q$ of $G$ associated with $\varphi$. When $\varphi$ is an involution, we describe its fixed point group $H=G^{\varphi}$ along with the $H$-orbits and $G$-orbits of $Q$ corresponding to the action of $\varphi$-twisted conjugation.