Statements (49)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:group_of_people
simple group permutation group |
| gptkbp:automorphismGroup |
trivial for n ≠ 6
S_6 for n=6 |
| gptkbp:centralTo |
trivial group
|
| gptkbp:definedIn |
group of even permutations on n elements
|
| gptkbp:firstNontrivialCase |
gptkb:A_3
|
| gptkbp:for_n=1 |
trivial group
|
| gptkbp:for_n=1_or_2 |
trivial group
|
| gptkbp:for_n=10 |
order 1814400
|
| gptkbp:for_n=2 |
trivial group
|
| gptkbp:for_n=3 |
cyclic group of order 3
isomorphic to cyclic group of order 3 |
| gptkbp:for_n=4 |
not simple
order 12 Klein four-group as normal subgroup |
| gptkbp:for_n=5 |
isomorphic to icosahedral group
smallest non-abelian simple group order 60 |
| gptkbp:for_n=6 |
order 360
|
| gptkbp:for_n=7 |
order 2520
|
| gptkbp:for_n=8 |
order 20160
|
| gptkbp:for_n=9 |
order 181440
|
| gptkbp:generation |
3-cycles
|
| gptkbp:hasMaximalSubgroup |
S_n for n > 4
|
| gptkbp:hasNormalSubgroup |
S_n (symmetric group)
|
| gptkbp:hasSubgroup |
S_n (symmetric group)
|
| https://www.w3.org/2000/01/rdf-schema#label |
A n (alternating group)
|
| gptkbp:identityElement |
identity permutation
|
| gptkbp:isNonAbelian |
false
true for n >= 5 |
| gptkbp:isomorphicTo |
PSL(2,5) for n=5
|
| gptkbp:isPerfect |
true for n >= 5
|
| gptkbp:isPrimitive |
true for n > 2
|
| gptkbp:isQuotientOf |
S_n by subgroup generated by a transposition
|
| gptkbp:isSimple |
n >= 5
|
| gptkbp:isTransitiveOn |
true
|
| gptkbp:minimalDegree |
n
|
| gptkbp:namedAfter |
alternating permutations
|
| gptkbp:notation |
A_n
|
| gptkbp:order |
n!/2
|
| gptkbp:usedIn |
gptkb:algebra
gptkb:mathematics gptkb:Galois_theory group theory combinatorics |
| gptkbp:bfsParent |
gptkb:S_n
|
| gptkbp:bfsLayer |
5
|