اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the Structure of Involutions and Symmetric Spaces of Dihedral Groups
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We initiate the study of analogues of symmetric spaces for the family of finite dihedral groups. In particular, we investigate the structure of the automorphism group, characterize the involutions of the automorphism group, and determine the fixed-group and symmetric space of each automorphism.
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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.
An involutive diffeomorphism $sigma$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quant
um field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs $(M,sigma)$, where $M$ is an irreducible symmetric space, not necessarily Riemannian, and $sigma$ is a dissecting involutive automorphism. In particular, we show that the only irreducible $1$-connected Riemannian symmetric spaces are $S^n$ and $H^n$ with dissecting isometric involutions whose fixed point spaces are $S^{n-1}$ and $H^{n-1}$, respectively.
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.
We consider groups of finite Morley rank with solvable local subgroups of even and mixed types. We also consider miscellaneous aspects of small groups of finite Morley rank of odd type.
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