Geometry & Groups
نویسنده
چکیده
According to Felix Klein’s influential Erlanger program of 1872, geometry is the study of properties of a space, which are invariant under a group of transformations. In Klein’s framework, the familiar Euclidean geometry consist of n-dimensional Euclidean space and its group of isometries. In general, a geometry is a pair (G,X), where X is a (sufficiently nice) space and G is a (sufficiently nice) group acting on the space. Geometric properties are precisely those that are preserved by the group. A geometry in Klein’s sense may not allow the concepts of distance or angle; an example of this is affine geometry.
منابع مشابه
Triple factorization of non-abelian groups by two maximal subgroups
The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...
متن کاملEffects of Hip Geometry on Fracture Patterns of Proximal Femur
Background: Some studies have previously shown that geometry of proximal femur can affect the probability of fracture and type of fracture. It happens since the geometry of the proximal femur determines how a force is applied to its different parts. In this study, we have compared proximal femur’s geometric characteristics in femoral neck (FNF), intertrochanteric (ITF) and Subtrochanteric (ST...
متن کاملDFT and HF Studies: Geometry, Hydrogen Bonding, Vibrational Frequencies and Electronic Properties of Enaminones and Their Complexes with Transition Metals
Enaminones are those structures made up three various functional groups including carbonyl, alkeneand amine groups which arelocated along with each other in a conjugate fashion. These compoundsare of much attention due to special characteristics and numerous applications. In the paper, sixvarious enaminone structures were theoretically optimized and after concluding, were compared withequivalen...
متن کاملAffine Geometry, Projective Geometry, and Non- Euclidean Geometry
1. Affine Geometry 1.1. Affine Space 1.2. Affine Lines 1.3. Affine transformations 1.4. Affine Collinearity 1.5. Conic Sections 2. Projective Geometry 2.1. Perspective 2.2. Projective Plane 2.3. Projective Transformations 2.4. Projective Collinearity 2.5. Conics 3. Geometries and Groups 3.1. Transformation Groups 3.2. Erlangen Program 4. Non-Euclidean Geometry 4.1. Elliptic Geometry 4.2. Hyperb...
متن کاملThe Alternating Group of Degree 6 in Geometry of the Leech Lattice and K3 Surfaces
The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a complex algebraic geometry of K3 surfaces.
متن کاملCannon-Thurston Maps and Kleinian Groups: Amalgamation Geometry and the 5-holed Sphere
We introduce the notion of amalgamation geometry manifolds. We show that the limit set of any surface group of amalgamated geometry is locally connected, thus giving a partial answer to a question (conjecture) raised by Cannon and Thurston, special cases of which have been obtained by Cannon and Thurston, Minsky, Klarreich and the author. The notion of amalgamated geometry includes, in a sense,...
متن کامل