Exemple #1
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def test_center():
    # the center of the dihedral group D_n is of order 2 for even n
    for i in (4, 6, 10):
        D = DihedralGroup(i)
        assert (D.center()).order() == 2
    # the center of the dihedral group D_n is of order 1 for odd n>2
    for i in (3, 5, 7):
        D = DihedralGroup(i)
        assert (D.center()).order() == 1
    # the center of an abelian group is the group itself
    for i in (2, 3, 5):
        for j in (1, 5, 7):
            for k in (1, 1, 11):
                G = AbelianGroup(i, j, k)
                assert G.center().is_subgroup(G)
    # the center of a nonabelian simple group is trivial
    for i in(1, 5, 9):
        A = AlternatingGroup(i)
        assert (A.center()).order() == 1
    # brute-force verifications
    D = DihedralGroup(5)
    A = AlternatingGroup(3)
    C = CyclicGroup(4)
    G.is_subgroup(D*A*C)
    assert _verify_centralizer(G, G)
Exemple #2
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def test_is_nilpotent():
    # every abelian group is nilpotent
    for i in (1, 2, 3):
        C = CyclicGroup(i)
        Ab = AbelianGroup(i, i + 2)
        assert C.is_nilpotent
        assert Ab.is_nilpotent
    Ab = AbelianGroup(5, 7, 10)
    assert Ab.is_nilpotent
    # A_5 is not solvable and thus not nilpotent
    assert AlternatingGroup(5).is_nilpotent is False
Exemple #3
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def test_cyclic():
    G = SymmetricGroup(2)
    assert G.is_cyclic
    G = AbelianGroup(3, 7)
    assert G.is_cyclic
    G = AbelianGroup(7, 7)
    assert not G.is_cyclic
    G = AlternatingGroup(3)
    assert G.is_cyclic
    G = AlternatingGroup(4)
    assert not G.is_cyclic
Exemple #4
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def test_cyclic():
    G = SymmetricGroup(2)
    assert G.is_cyclic
    G = AbelianGroup(3, 7)
    assert G.is_cyclic
    G = AbelianGroup(7, 7)
    assert not G.is_cyclic
    G = AlternatingGroup(3)
    assert G.is_cyclic
    G = AlternatingGroup(4)
    assert not G.is_cyclic

    # Order less than 6
    G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 2, 1))
    assert G.is_cyclic
    G = PermutationGroup(
        Permutation(0, 1, 2, 3),
        Permutation(0, 2)(1, 3)
    )
    assert G.is_cyclic
    G = PermutationGroup(
        Permutation(3),
        Permutation(0, 1)(2, 3),
        Permutation(0, 2)(1, 3),
        Permutation(0, 3)(1, 2)
    )
    assert G.is_cyclic is False

    # Order 15
    G = PermutationGroup(
        Permutation(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14),
        Permutation(0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13)
    )
    assert G.is_cyclic

    # Distinct prime orders
    assert PermutationGroup._distinct_primes_lemma([3, 5]) is True
    assert PermutationGroup._distinct_primes_lemma([5, 7]) is True
    assert PermutationGroup._distinct_primes_lemma([2, 3]) is None
    assert PermutationGroup._distinct_primes_lemma([3, 5, 7]) is None
    assert PermutationGroup._distinct_primes_lemma([5, 7, 13]) is True

    G = PermutationGroup(
        Permutation(0, 1, 2, 3),
        Permutation(0, 2)(1, 3))
    assert G.is_cyclic
    assert G._is_abelian
Exemple #5
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def test_abelian_invariants():
    G = AbelianGroup(2, 3, 4)
    assert G.abelian_invariants() == [2, 3, 4]
    G = PermutationGroup(
        [Permutation(1, 2, 3, 4),
         Permutation(1, 2),
         Permutation(5, 6)])
    assert G.abelian_invariants() == [2, 2]
    G = AlternatingGroup(7)
    assert G.abelian_invariants() == []
    G = AlternatingGroup(4)
    assert G.abelian_invariants() == [3]
    G = DihedralGroup(4)
    assert G.abelian_invariants() == [2, 2]

    G = PermutationGroup([Permutation(1, 2, 3, 4, 5, 6, 7)])
    assert G.abelian_invariants() == [7]
    G = DihedralGroup(12)
    S = G.sylow_subgroup(3)
    assert S.abelian_invariants() == [3]
    G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 2, 3))
    assert G.abelian_invariants() == [3]
    G = PermutationGroup(
        [Permutation(0, 1),
         Permutation(0, 2, 4, 6)(1, 3, 5, 7)])
    assert G.abelian_invariants() == [2, 4]
    G = SymmetricGroup(30)
    S = G.sylow_subgroup(2)
    assert S.abelian_invariants() == [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
    S = G.sylow_subgroup(3)
    assert S.abelian_invariants() == [3, 3, 3, 3]
    S = G.sylow_subgroup(5)
    assert S.abelian_invariants() == [5, 5, 5]
def test_AbelianGroup():
    A = AbelianGroup(3, 3, 3)
    assert A.order() == 27
    assert A.is_abelian == True
Exemple #7
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def test_AbelianGroup():
    A = AbelianGroup(3, 3, 3)
    assert A.order() == 27
    assert A.is_abelian is True