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)
def test_commutator():
# the commutator of the trivial group and the trivial group is trivial
S = SymmetricGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert S.commutator(triv, triv).is_subgroup(triv)
# the commutator of the trivial group and any other group is again trivial
A = AlternatingGroup(3)
assert S.commutator(triv, A).is_subgroup(triv)
# the commutator is commutative
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
assert S.commutator(A, D).is_subgroup(S.commutator(D, A))
# the commutator of an abelian group is trivial
S = SymmetricGroup(7)
A1 = AbelianGroup(2, 5)
A2 = AbelianGroup(3, 4)
triv = PermutationGroup([Permutation([0, 1, 2, 3, 4, 5, 6])])
assert S.commutator(A1, A1).is_subgroup(triv)
assert S.commutator(A2, A2).is_subgroup(triv)
# examples calculated by hand
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert S.commutator(A, S).is_subgroup(A)
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
def test_Permutation_Cycle():
# general principle: economically, canonically show all moved elements
# and the size of the permutation.
for p, s in [
(Cycle(), 'Cycle()'),
(Cycle(2), 'Cycle(2)'),
(Cycle(2, 1), 'Cycle(1, 2)'),
(Cycle(1, 2)(5)(6, 7)(10), 'Cycle(1, 2)(6, 7)(10)'),
(Cycle(3, 4)(1, 2)(3, 4), 'Cycle(1, 2)(4)'),
]:
assert str(p) == s
Permutation.print_cyclic = False
for p, s in [
(Permutation([]), 'Permutation([])'),
(Permutation([], size=1), 'Permutation([0])'),
(Permutation([], size=2), 'Permutation([0, 1])'),
(Permutation([], size=10), 'Permutation([], size=10)'),
(Permutation([1, 0, 2]), 'Permutation([1, 0, 2])'),
(Permutation([1, 0, 2, 3, 4, 5]), 'Permutation([1, 0], size=6)'),
(Permutation([1, 0, 2, 3, 4, 5],
size=10), 'Permutation([1, 0], size=10)'),
]:
assert str(p) == s
Permutation.print_cyclic = True
for p, s in [
(Permutation([]), 'Permutation()'),
(Permutation([], size=1), 'Permutation(0)'),
(Permutation([], size=2), 'Permutation(1)'),
(Permutation([], size=10), 'Permutation(9)'),
(Permutation([1, 0, 2]), 'Permutation(2)(0, 1)'),
(Permutation([1, 0, 2, 3, 4, 5]), 'Permutation(5)(0, 1)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10), 'Permutation(9)(0, 1)'),
(Permutation([0, 1, 3, 2, 4, 5], size=10), 'Permutation(9)(2, 3)'),
]:
assert str(p) == s
assert str(AbelianGroup(3, 4)) == ("PermutationGroup([\n "
"Permutation(6)(0, 1, 2),\n"
" Permutation(3, 4, 5, 6)])")
assert sstr(Cycle(1, 2)) == repr(Cycle(1, 2))
def test_AbelianGroup():
A = AbelianGroup(3, 3, 3)
assert A.order() == 27
assert A.is_abelian is True