Statements (61)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:musical_group
|
| gptkbp:application |
in physics
in algebra in geometry in combinatorics |
| gptkbp:characteristic |
has a normal subgroup of index 2
|
| gptkbp:contains |
the alternating group A_n
|
| gptkbp:defines |
the group of all permutations of a finite set
|
| gptkbp:denoted_by |
gptkb:S_n
|
| gptkbp:has |
a group S_3 which is the smallest non-abelian group
a Cayley graph representation a group S_2 which is isomorphic to Z/2 Z a group S_4 which has 24 elements a group S_5 which has 120 elements a representation in terms of Young tableaux a representation in terms of character theory a representation in terms of group actions a representation in terms of permutation matrices a trivial group S_1 n! elements a group S_n which is the symmetric group on n letters |
| gptkbp:has_produced |
transpositions
|
| gptkbp:has_subgroup |
the symmetric group on a larger set
permutation group |
| https://www.w3.org/2000/01/rdf-schema#label |
symmetric groups
|
| gptkbp:is_related_to |
the concept of symmetry
the concept of bijections the concept of normal subgroups the concept of orbits the concept of Schur functions the concept of automorphisms the concept of conjugacy classes the concept of cosets the concept of finite groups the concept of group actions on sets the concept of group actions on vector spaces the concept of group cohomology the concept of group extensions the concept of group representations the concept of homomorphisms the concept of infinite groups the concept of isomorphisms the concept of linear representations the concept of nilpotent groups the concept of projective representations the concept of representation theory the concept of simple groups the concept of solvable groups the concept of stabilizers the concept of symmetric functions the concept of symmetric polynomials |
| gptkbp:is_used_in |
the study of algebraic structures
the study of combinatorial designs the study of graph theory the study of Galois theory the study of coding theory |
| gptkbp:isomorphic_to |
the full symmetric group on n letters
|
| gptkbp:notable_traits |
non-abelian for n > 2
|
| gptkbp:order |
n! (n factorial)
|
| gptkbp:bfsParent |
gptkb:Burnside's_theorem
|
| gptkbp:bfsLayer |
6
|