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gptkbp:instanceOf
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gptkb:mathematical_concept
group theory concept
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gptkbp:A_1
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trivial group
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gptkbp:A_2
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trivial group
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gptkbp:A_3
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cyclic group of order 3
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gptkbp:A_3_is_abelian
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true
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gptkbp:A_3_isomorphic_to
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cyclic group of order 3
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gptkbp:A_4
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group of order 12
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gptkbp:A_4_is_abelian
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false
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gptkbp:A_4_is_solvable
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true
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gptkbp:A_4_isomorphic_to
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group of even permutations on 4 elements
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gptkbp:A_5
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group of order 60
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gptkbp:A_5_is_abelian
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false
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gptkbp:A_5_is_not_solvable
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true
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gptkbp:A_5_is_the_smallest_non-abelian_simple_group
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true
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gptkbp:A_5_isomorphic_to
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gptkb:icosahedral_group
simple group of order 60
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gptkbp:A_n_is_a_normal_subgroup_of_S_n
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true
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gptkbp:A_n_is_a_subgroup_of_S_n
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true
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gptkbp:A_n_is_doubly_transitive
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for n ≥ 4
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gptkbp:A_n_is_generated_by_3-cycles
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true
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gptkbp:A_n_is_index_2_subgroup_of_S_n
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true
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gptkbp:A_n_is_non-abelian
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for n ≥ 4
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gptkbp:A_n_is_not_simple
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for n < 5
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gptkbp:A_n_is_not_solvable
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for n ≥ 5
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gptkbp:A_n_is_primitive
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for n ≥ 5
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gptkbp:A_n_is_simple
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for n ≥ 5
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gptkbp:A_n_is_the_commutator_subgroup_of_S_n
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for n ≥ 5
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gptkbp:A_n_is_the_kernel_of_the_sign_homomorphism
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from S_n to {1, -1}
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gptkbp:A_n_is_transitive
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true
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gptkbp:A_n_is_used_in
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gptkb:Galois_theory
classification of finite simple groups
algebraic equations
permutation group theory
solvability of polynomials
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gptkbp:defines
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group of even permutations of n elements
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gptkbp:hasNormalSubgroup
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gptkb:symmetric_group_S_n
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https://www.w3.org/2000/01/rdf-schema#label
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alternating groups
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gptkbp:isA
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gptkb:group_of_people
simple group (for n ≥ 5)
subgroup of symmetric group
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gptkbp:isNonAbelian
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false (for n ≥ 4)
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gptkbp:isSimple
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true (for n ≥ 5)
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gptkbp:notation
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A_n
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gptkbp:order
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n!/2
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gptkbp:bfsParent
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gptkb:sporadic_groups
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gptkbp:bfsLayer
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6
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