def test_verify_centralizer():
# verified by GAP
S = SymmetricGroup(3)
A = AlternatingGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert _verify_centralizer(S, S, centr=triv)
assert _verify_centralizer(S, A, centr=A)
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_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_centralizer():
# the centralizer of the trivial group is the entire group
S = SymmetricGroup(2)
assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S)
A = AlternatingGroup(5)
assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A)
# a centralizer in the trivial group is the trivial group itself
triv = PermutationGroup([Permutation([0, 1, 2, 3])])
D = DihedralGroup(4)
assert triv.centralizer(D).is_subgroup(triv)
# brute-force verifications for centralizers of groups
for i in (4, 5, 6):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
D = DihedralGroup(i)
for gp in (S, A, C, D):
for gp2 in (S, A, C, D):
if not gp2.is_subgroup(gp):
assert _verify_centralizer(gp, gp2)
# verify the centralizer for all elements of several groups
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_centralizer(S, element)
A = AlternatingGroup(5)
elements = list(A.generate_dimino())
for element in elements:
assert _verify_centralizer(A, element)
D = DihedralGroup(7)
elements = list(D.generate_dimino())
for element in elements:
assert _verify_centralizer(D, element)
# verify centralizers of small groups within small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp.degree == gp2.degree:
assert _verify_centralizer(gp, gp2)
def test_centralizer():
# the centralizer of the trivial group is the entire group
S = SymmetricGroup(2)
assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S)
A = AlternatingGroup(5)
assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A)
# a centralizer in the trivial group is the trivial group itself
triv = PermutationGroup([Permutation([0, 1, 2, 3])])
D = DihedralGroup(4)
assert triv.centralizer(D).is_subgroup(triv)
# brute-force verifications for centralizers of groups
for i in (4, 5, 6):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
D = DihedralGroup(i)
for gp in (S, A, C, D):
for gp2 in (S, A, C, D):
if not gp2.is_subgroup(gp):
assert _verify_centralizer(gp, gp2)
# verify the centralizer for all elements of several groups
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_centralizer(S, element)
A = AlternatingGroup(5)
elements = list(A.generate_dimino())
for element in elements:
assert _verify_centralizer(A, element)
D = DihedralGroup(7)
elements = list(D.generate_dimino())
for element in elements:
assert _verify_centralizer(D, element)
# verify centralizers of small groups within small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp.degree == gp2.degree:
assert _verify_centralizer(gp, gp2)