def automorphism_group(self):
"""
Returns the subgroup of the automorphism group of the incidence
graph which respects the P B partition. This is (isomorphic to) the
automorphism group of the block design, although the degrees
differ.
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: G = BD.automorphism_group(); G
Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]
sage: BD = BlockDesign(4,[[0],[0,1],[1,2],[3,3]],test=False)
sage: G = BD.automorphism_group(); G
Permutation Group with generators [()]
sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: G = BD.automorphism_group(); G
Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]
"""
from sage.groups.perm_gps.partn_ref.refinement_matrices import MatrixStruct
from sage.groups.perm_gps.permgroup import PermutationGroup
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
M1 = self.incidence_matrix()
M2 = MatrixStruct(M1)
M2.run()
gens = M2.automorphism_group()[0]
v = len(self.points())
G = SymmetricGroup(v)
gns = []
for g in gens:
L = [j+1 for j in g]
gns.append(G(L))
return PermutationGroup(gns)
def automorphism_group(self):
"""
Returns the subgroup of the automorphism group of the incidence
graph which respects the P B partition. This is (isomorphic to) the
automorphism group of the block design, although the degrees
differ.
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: G = BD.automorphism_group(); G
Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]
sage: BD = BlockDesign(4,[[0],[0,1],[1,2],[3,3]],test=False)
sage: G = BD.automorphism_group(); G
Permutation Group with generators [()]
sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: G = BD.automorphism_group(); G
Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]
"""
from sage.groups.perm_gps.partn_ref.refinement_matrices import MatrixStruct
from sage.groups.perm_gps.permgroup import PermutationGroup
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
M1 = self.incidence_matrix()
M2 = MatrixStruct(M1)
M2.run()
gens = M2.automorphism_group()[0]
v = len(self.points())
G = SymmetricGroup(v)
gns = []
for g in gens:
L = [j+1 for j in g]
gns.append(G(L))
return PermutationGroup(gns)
def automorphism_group(self):
"""
Returns the subgroup of the automorphism group of the incidence graph
which respects the P B partition. It is (isomorphic to) the automorphism
group of the block design, although the degrees differ.
EXAMPLES::
sage: P = designs.DesarguesianProjectivePlaneDesign(2); P
Incidence structure with 7 points and 7 blocks
sage: G = P.automorphism_group()
sage: G.is_isomorphic(PGL(3,2))
True
sage: G
Permutation Group with generators [(2,3)(4,5), (2,4)(3,5), (1,2)(4,6), (0,1)(4,5)]
A non self-dual example::
sage: from sage.combinat.designs.incidence_structures import IncidenceStructure
sage: IS = IncidenceStructure(range(4), [[0,1,2,3],[1,2,3]])
sage: IS.automorphism_group().cardinality()
6
sage: IS.dual_design().automorphism_group().cardinality()
1
"""
from sage.groups.perm_gps.partn_ref.refinement_matrices import MatrixStruct
from sage.groups.perm_gps.permgroup import PermutationGroup
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
M1 = self.incidence_matrix().transpose()
M2 = MatrixStruct(M1)
M2.run()
gens = M2.automorphism_group()[0]
return PermutationGroup(gens, domain=range(self.v))
def automorphism_group(self):
r"""
Return the subgroup of the automorphism group of the incidence graph
which respects the P B partition. It is (isomorphic to) the automorphism
group of the block design, although the degrees differ.
EXAMPLES::
sage: P = designs.DesarguesianProjectivePlaneDesign(2); P
Incidence structure with 7 points and 7 blocks
sage: G = P.automorphism_group()
sage: G.is_isomorphic(PGL(3,2))
True
sage: G
Permutation Group with generators [(2,3)(4,5), (2,4)(3,5), (1,2)(4,6), (0,1)(4,5)]
A non self-dual example::
sage: IS = designs.IncidenceStructure(range(4), [[0,1,2,3],[1,2,3]])
sage: IS.automorphism_group().cardinality()
6
sage: IS.dual().automorphism_group().cardinality()
1
Examples with non-integer points::
sage: I = designs.IncidenceStructure('abc', ('ab','ac','bc'))
sage: I.automorphism_group()
Permutation Group with generators [('b','c'), ('a','b')]
sage: designs.IncidenceStructure([[(1,2),(3,4)]]).automorphism_group()
Permutation Group with generators [((1,2),(3,4))]
"""
from sage.groups.perm_gps.partn_ref.refinement_matrices import MatrixStruct
from sage.groups.perm_gps.permgroup import PermutationGroup
from sage.groups.perm_gps.permgroup_element import standardize_generator
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
M1 = self.incidence_matrix().transpose()
M2 = MatrixStruct(M1)
M2.run()
gens = M2.automorphism_group()[0]
gens = [standardize_generator([x + 1 for x in g]) for g in gens]
if self._point_to_index:
gens = [[
tuple([self._points[i - 1] for i in cycle]) for cycle in g
] for g in gens]
else:
gens = [[tuple([i - 1 for i in cycle]) for cycle in g]
for g in gens]
return PermutationGroup(gens, domain=self._points)
def automorphism_group(self):
r"""
Return the subgroup of the automorphism group of the incidence graph
which respects the P B partition. It is (isomorphic to) the automorphism
group of the block design, although the degrees differ.
EXAMPLES::
sage: P = designs.DesarguesianProjectivePlaneDesign(2); P
Incidence structure with 7 points and 7 blocks
sage: G = P.automorphism_group()
sage: G.is_isomorphic(PGL(3,2))
True
sage: G
Permutation Group with generators [(2,3)(4,5), (2,4)(3,5), (1,2)(4,6), (0,1)(4,5)]
A non self-dual example::
sage: IS = designs.IncidenceStructure(range(4), [[0,1,2,3],[1,2,3]])
sage: IS.automorphism_group().cardinality()
6
sage: IS.dual().automorphism_group().cardinality()
1
Examples with non-integer points::
sage: I = designs.IncidenceStructure('abc', ('ab','ac','bc'))
sage: I.automorphism_group()
Permutation Group with generators [('b','c'), ('a','b')]
sage: designs.IncidenceStructure([[(1,2),(3,4)]]).automorphism_group()
Permutation Group with generators [((1,2),(3,4))]
"""
from sage.groups.perm_gps.partn_ref.refinement_matrices import MatrixStruct
from sage.groups.perm_gps.permgroup import PermutationGroup
from sage.groups.perm_gps.permgroup_element import standardize_generator
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
M1 = self.incidence_matrix().transpose()
M2 = MatrixStruct(M1)
M2.run()
gens = M2.automorphism_group()[0]
gens = [standardize_generator([x+1 for x in g]) for g in gens]
if self._point_to_index:
gens = [[tuple([self._points[i-1] for i in cycle]) for cycle in g] for g in gens]
else:
gens = [[tuple([i-1 for i in cycle]) for cycle in g] for g in gens]
return PermutationGroup(gens, domain=self._points)