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 $H_{theta}=G^{theta}={xin G mid theta(x)=x}$ and generalized symmetric space $Q$
of $theta$ given by $Q_{theta}={gin G mid g=xtheta(x)^{-1}~mbox{for some}~xin G}.$ The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,~~H.Q eq QD_{8k}.$ 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 $R=R_{theta}={~xin G~mid~theta(x)=x^{-1}}$ has been explored.
We propose a new class of mathematical structures called (m,n)-semirings} (which generalize the usual semirings), and describe their basic properties. We also define partial ordering, and generalize the concepts of congruence, homomorphism, ideals, e
tc., for (m,n)-semirings. Following earlier work by Rao, we consider a system as made up of several components whose failures may cause it to fail, and represent the set of systems algebraically as an (m,n)-semiring. Based on the characteristics of these components we present a formalism to compare the fault tolerance behaviour of two systems using our framework of a partially ordered (m,n)-semiring.
