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gptkbp:instanceOf
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gptkb:group_of_people
gptkb:permutation_group
gptkb:simple_group
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gptkbp:automorphismGroup
|
trivial for n ≠ 6
S_6 for n=6
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gptkbp:centralTo
|
trivial group
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gptkbp:definedIn
|
group of even permutations on n elements
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gptkbp:firstNontrivialCase
|
gptkb:A_3
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gptkbp:for_n=1
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trivial group
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gptkbp:for_n=1_or_2
|
trivial group
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gptkbp:for_n=10
|
order 1814400
|
|
gptkbp:for_n=2
|
trivial group
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|
gptkbp:for_n=3
|
cyclic group of order 3
isomorphic to cyclic group of order 3
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|
gptkbp:for_n=4
|
not simple
order 12
Klein four-group as normal subgroup
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|
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)
|
|
gptkbp:identityElement
|
identity permutation
|
|
gptkbp:isNonAbelian
|
false
true for n >= 5
|
|
gptkbp:isomorphicTo
|
PSL(2,5) for n=5
|
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gptkbp:isPerfect
|
true for n >= 5
|
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gptkbp:isPrimitive
|
true for n > 2
|
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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:combinatorics
gptkb:algebra
gptkb:mathematics
gptkb:Galois_theory
group theory
|
|
gptkbp:bfsParent
|
gptkb:S_n
|
|
gptkbp:bfsLayer
|
9
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
A n (alternating group)
|