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  2. Center (group theory) - Wikipedia

    en.wikipedia.org/wiki/Center_(group_theory)

    The kernel of the map G → G i is the i th center [1] of G (second center, third center, etc.), denoted Z i (G). [2] Concretely, the (i+1)-st center comprises the elements that commute with all elements up to an element of the i th center. Following this definition, one can define the 0th center of a group to be the identity subgroup.

  3. List of spherical symmetry groups - Wikipedia

    en.wikipedia.org/wiki/List_of_spherical_symmetry...

    The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. [ 3 ] Hermann–Mauguin notation (International notation) is also given.

  4. Symmetry - Wikipedia

    en.wikipedia.org/wiki/Symmetry

    This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art, and music. The opposite of symmetry is asymmetry, which refers to the absence of symmetry.

  5. Symmetry group - Wikipedia

    en.wikipedia.org/wiki/Symmetry_group

    In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.

  6. Symmetry (geometry) - Wikipedia

    en.wikipedia.org/wiki/Symmetry_(geometry)

    A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are: The group of similarity transformations; [30] i.e., affine transformations represented by a matrix A that is a scalar times an orthogonal matrix.

  7. Symmetric group - Wikipedia

    en.wikipedia.org/wiki/Symmetric_group

    In the theory of Coxeter groups, the symmetric group is the Coxeter group of type A n and occurs as the Weyl group of the general linear group. In combinatorics , the symmetric groups, their elements ( permutations ), and their representations provide a rich source of problems involving Young tableaux , plactic monoids , and the Bruhat order .

  8. Centrosymmetry - Wikipedia

    en.wikipedia.org/wiki/Centrosymmetry

    In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. [1] In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have inversion symmetry. [2] Point reflection is a similar term used in geometry.

  9. Geometric group theory - Wikipedia

    en.wikipedia.org/wiki/Geometric_group_theory

    Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...

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