On the Structure of Involutions and Symmetric Spaces of Quasi Dihedral Group

Let $G=QD_{8k}~$ be the quasi-dihedral group of order $8n$ and $\theta$ be an automorphism of $QD_{8k}$ of finite order. The fixed-point set $H$ of $\theta$ is defined as $H_{\theta}=G^{\theta}=\{x\in G \mid \theta(x)=x\}$ and generalized symmetric space $Q$ of $\theta$ given by $Q_{\theta}=\{g\in G \mid g=x\theta(x)^{-1}~\mbox{for some}~x\in G\}.$ The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,~~H.Q\neq QD_{8k}.$ the $H$-orbits on $Q$ are obtained under different conditions. Moreover, the formula to find the order of $v$-th root of unity in $\mathbb{Z}_{2k}$ for $QD_{8k}$ has been calculated. The criteria to find the number of equivalence classes denoted by $C_{4k}$ of the involution automorphism has also been constructed. Finally, the set of twisted involutions $R=R_{\theta}=\{~x\in G~\mid~\theta(x)=x^{-1}\}$ has been explored.