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  2. Symmetric group - Wikipedia

    en.wikipedia.org/wiki/Symmetric_group

    A Cayley graph of the symmetric group S 4 using the generators (red) a right circular shift of all four set elements, and (blue) a left circular shift of the first three set elements. Cayley table, with header omitted, of the symmetric group S 3. The elements are represented as matrices. To the left of the matrices, are their two-line form.

  3. List of character tables for chemically important 3D point ...

    en.wikipedia.org/wiki/List_of_character_tables...

    The finite group notation used is: Z n: cyclic group of order n, D n: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, S n: symmetric group on n letters, and A n: alternating group on n letters. The character tables then follow for all groups. The rows of the character tables correspond to the irreducible ...

  4. Dihedral group of order 6 - Wikipedia

    en.wikipedia.org/wiki/Dihedral_group_of_order_6

    Only the neutral elements are symmetric to the main diagonal, so this group is not abelian. Cayley table as general (and special) linear group GL(2, 2) In mathematics, D 3 (sometimes alternatively denoted by D 6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3. It is also the smallest non-abelian group. [1] This ...

  5. Point groups in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Point_groups_in_three...

    This group has six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single S 4 axis, and two C 3 axes. T d is isomorphic to S 4, the symmetric group on 4 letters, because there is a 1-to-1 correspondence between the elements of T d and the 24 permutations of the four 3-fold

  6. Automorphisms of the symmetric and alternating groups

    en.wikipedia.org/wiki/Automorphisms_of_the...

    For every symmetric group other than S 6, there is no other conjugacy class consisting of elements of order 2 that has the same number of elements as the class of transpositions. Or as follows: Each permutation of order two (called an involution ) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2 k 1 n −2 k .

  7. SQL syntax - Wikipedia

    en.wikipedia.org/wiki/SQL_syntax

    The SQL language is subdivided into several language elements, including: Keywords are words that are defined in the SQL language. They are either reserved (e.g. SELECT, COUNT and YEAR), or non-reserved (e.g. ASC, DOMAIN and KEY). List of SQL reserved words. Identifiers are names on database objects, like tables, columns and schemas. An ...

  8. Clebsch–Gordan coefficients for SU(3) - Wikipedia

    en.wikipedia.org/wiki/Clebsch–Gordan...

    The permutations of n identical particles constitute the symmetric group S n. Every n-particle state of S n that is made up of single-particle states of the fundamental N-dimensional SU(N) multiplet belongs to an irreducible SU(N) representation. Thus, it can be used to determine the Clebsch–Gordan series for any unitary group. [17]

  9. Centralizer and normalizer - Wikipedia

    en.wikipedia.org/wiki/Centralizer_and_normalizer

    In fact, C G (S) is always a normal subgroup of N G (S), being the kernel of the homomorphism N G (S) → Bij(S) and the group N G (S)/C G (S) acts by conjugation as a group of bijections on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as W ( G , T ) = N G ( T )/C G ( T ) , and especially if the torus is maximal (i ...