symmetric group S n (n ≥ 3)

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:group_of_people gptkb:permutation_group
gptkbp:actsOn	set of n elements
gptkbp:application	gptkb:combinatorics gptkb:algebra gptkb:geometry gptkb:Galois_theory physics
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)
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
https://www.w3.org/2000/01/rdf-schema#label	symmetric group S n (n ≥ 3)