Statements (43)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:group_of_people
permutation group |
| gptkbp:actsOn |
set of n elements
|
| gptkbp:application |
gptkb:algebra
gptkb:geometry gptkb:Galois_theory physics combinatorics |
| gptkbp:automorphismGroup |
order 2 (for n = 6)
trivial (for n ≠ 6) S_n (for n ≠ 6) |
| gptkbp:CayleyTable |
available
|
| gptkbp:centralTo |
trivial group
|
| gptkbp:conjugacyClasses |
indexed by cycle type
|
| gptkbp:contains |
identity permutation
all possible permutations of n elements even and odd permutations n! elements |
| gptkbp:generation |
transpositions
|
| gptkbp:hasElementOrder |
k for 1 ≤ k ≤ n
|
| gptkbp:hasMaximalSubgroup |
A_n
S_{n-1} |
| gptkbp:hasNormalSubgroup |
gptkb:alternating_group_A_n
|
| gptkbp:hasSubgroup |
gptkb:alternating_group_A_n
|
| gptkbp:hasSubgroupOfEveryOrder |
true (for n ≥ 3)
|
| https://www.w3.org/2000/01/rdf-schema#label |
symmetric group S n (n ≥ 3)
|
| gptkbp:isNonAbelian |
true
|
| gptkbp:isomorphicTo |
automorphism group of n-element set
automorphism group of complete graph K_n |
| gptkbp:isPrimitive |
true
|
| gptkbp:isQuotientOf |
A_n gives C_2 (for n ≥ 2)
|
| gptkbp:isSimple |
false
|
| gptkbp:isSolvable |
false (for n ≥ 5)
true (for n = 3, 4) |
| gptkbp:isTransitiveOn |
true
|
| gptkbp:isUniversal |
every finite group is isomorphic to a subgroup of some S_n (Cayley's theorem)
|
| gptkbp:namedAfter |
permutations (symmetry)
|
| gptkbp:notation |
gptkb:S_n
|
| gptkbp:order |
n!
|
| gptkbp:presentedBy |
generated by (1 2), (1 2 ... n) with relations
|
| gptkbp:representationTheory |
irreducible representations correspond to partitions of n
|
| gptkbp:bfsParent |
gptkb:symmetric_group_S3
|
| gptkbp:bfsLayer |
7
|