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The center of the symmetric group, S n, is trivial for n ≥ 3. The center of the alternating group, A n, is trivial for n ≥ 4. The center of the general linear group over a field F, GL n (F), is the collection of scalar matrices, { sI n ∣ s ∈ F \ {0} }. The center of the orthogonal group, O n (F) is {I n, −I n}.
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 .
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.
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
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.
The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of all transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry.
In geometry, a centre (British English) or center (American English) (from Ancient Greek κέντρον (kéntron) 'pointy object') of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre.
In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group. Given a group G {\displaystyle G} , the center of G {\displaystyle G} , denoted as Z ( G ) {\displaystyle Z(G)} , is defined as the set of those elements of the group which commute with every element of the group.