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In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group ''E''(3) of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them. The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups. Finite Coxeter groups are a special set of ''point groups'' generated purely by a set of reflectional mirrors passing through the same point. A rank ''n'' Coxeter group has ''n'' mirrors and is represented by a Coxeter-Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. ==Group structure== SO(3) is a subgroup of ''E''+(3), which consists of ''direct isometries'', i.e., isometries preserving orientation; it contains those that leave the origin fixed. O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix −''I''): :O(3) = SO(3) × Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries ''H'' in O(3) and all groups ''K'' of isometries in O(3) that contain inversion: :''K'' = ''H'' × :''H'' = ''K'' ∩ SO(3) For instance, if ''H'' is ''C''2, then ''K'' is ''C''2h, or if ''H'' is ''C''3, then ''K'' is ''S''6. (See lower down for the definitions of these groups.) If a group of direct isometries ''H'' has a subgroup ''L'' of index 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion: :''M'' = ''L'' ∪ ( (''H'' \ ''L'') × ) where isometry ( ''A'', ''I'' ) is identified with ''A''. An example would be ''C''4 for ''H'' and ''S''4 for ''M''. Thus ''M'' is obtained from ''H'' by inverting the isometries in ''H'' \ ''L''. This group ''M'' is as abstract group isomorphic with ''H''. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below. In 2D the cyclic group of ''k''-fold rotations ''Ck'' is for every positive integer ''k'' a normal subgroup of O(2,R) and SO(2,R). Accordingly, in 3D, for every axis the cyclic group of ''k''-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations (''Cn'') is normal both in the group obtained by adding reflections in planes through the axis (''Cnv'') and in the group obtained by adding a reflection plane perpendicular to the axis (''Cnh''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Point groups in three dimensions」の詳細全文を読む スポンサード リンク
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