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In abstract algebra, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in more general contexts as expanded below. ==Introduction== The "objects" may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g., a function of position with values in a set of colors. For symmetry of physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a group action on objects in it. The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations, and compositions of these) that leave the figure invariant is called its proper symmetry group. The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant). Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of the orthogonal group O(''n'') by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(''n''), and is therefore also called rotation group of the figure. A discrete symmetry group is a symmetry group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversion and rotoinversion – they are just the finite subgroups of O(''n''), (2) infinite lattice groups, which include only translations, and (3) infinite space groups which combines elements of both previous types, and may also include extra transformations like screw axis and glide reflection. There are also ''continuous'' symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(''n'') (the isometry group of R''n''), where two subgroups ''H''1, ''H''2 of a group ''G'' are ''conjugate'', if there exists such that . For example: *two 3D figures have mirror symmetry, but with respect to different mirror planes. *two 3D figures have 3-fold rotational symmetry, but with respect to different axes. *two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also those involved in continuous symmetries, but excludes for example in 1D the group of translations by a rational number. A "figure" with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous. 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「Symmetry group」の詳細全文を読む スポンサード リンク
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