groups, the symmetric group is the Coxeter group of type An and occurs as the Weyl group of the general linear group. In combinatorics, the symmetric...

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of...

potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group Sn has order n!. Its conjugacy...

the alternating group on n letters and denoted by An or Alt(n). For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index...

and exactly two non-real roots, then the Galois group of f {\displaystyle f} is the full symmetric group S p . {\displaystyle S_{p}.} For example, f ( x...

i<j\leq n} . This is invariant under the symmetric group, and Y {\displaystyle Y} is the quotient by the symmetric group of the non-excluded n {\displaystyle...

dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group. This page illustrates many group concepts...

permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is...

a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself. Likewise, a right group action...

complex Lie algebra. Symmetric spaces are classified as follows. First, the universal cover of a symmetric space is still symmetric, so we can reduce to...

Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. The symmetric difference is equivalent...

and the symmetric group of order n. For m = 1 , {\displaystyle m=1,} the generalized symmetric group is exactly the ordinary symmetric group: S ( 1 ,...

four-group are interchangeable: the automorphism group of V is thus the group of permutations of these three elements, that is, the symmetric group S 3...

subgroups of its Galois group for expressing this characterization in terms of solvable groups; the proof that the symmetric group is not solvable if its...

equal). Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries...

In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples...

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically...

exist solvable groups that are not CLT (for example, A4) and CLT groups that are not supersolvable (for example, S4, the symmetric group of degree 4)....

outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group, Sn, conjugation...

that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S3) = 6, the possible...

PGL is the full symmetric group), PGL is a proper subgroup of the full symmetric group on these points. For n ≥ 3, the collineation group is the projective...

a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn; in general, any permutation group G...

In finite group theory, Jordan's theorem states that if a primitive permutation group G is a subgroup of the symmetric group Sn and contains a p-cycle...

for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic. S4 is the symmetric group whose elements...

represented by their respective permutations. The Rubik's Cube group is the subgroup of the symmetric group S 48 {\displaystyle S_{48}} generated by the six permutations...

right-hand side is a sum of characters for symmetric groups that have smaller order than that of the symmetric group we started with on the left-hand side...

non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of invertible linear...

Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries...

symplectic group. There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups: S3 ≅ PSL2(2) ≅ dihedral group of...

in a uniform way; for example, in the symmetric group they can be described by cycle type. The symmetric group S 3 , {\displaystyle S_{3},} consisting...