def low_index_subgroups(G, N, Y=[]):
"""
Implements the Low Index Subgroups algorithm, i.e find all subgroups of
``G`` upto a given index ``N``. This implements the method described in
[Sim94]. This procedure involves a backtrack search over incomplete Coset
Tables, rather than over forced coincidences.
Parameters
==========
G: An FpGroup < X|R >
N: positive integer, representing the maximum index value for subgroups
Y: (an optional argument) specifying a list of subgroup generators, such
that each of the resulting subgroup contains the subgroup generated by Y.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, low_index_subgroups
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**2, y**3, (x*y)**4])
>>> L = low_index_subgroups(f, 4)
>>> for coset_table in L:
... print(coset_table.table)
[[0, 0, 0, 0]]
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]]
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]]
[[1, 1, 0, 0], [0, 0, 1, 1]]
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 5.4
.. [2] Marston Conder and Peter Dobcsanyi
"Applications and Adaptions of the Low Index Subgroups Procedure"
"""
C = CosetTable(G, [])
R = G.relators
# length chosen for the length of the short relators
len_short_rel = 5
# elements of R2 only checked at the last step for complete
# coset tables
R2 = set([rel for rel in R if len(rel) > len_short_rel])
# elements of R1 are used in inner parts of the process to prune
# branches of the search tree,
R1 = set([rel.identity_cyclic_reduction() for rel in set(R) - R2])
R1_c_list = C.conjugates(R1)
S = []
descendant_subgroups(S, C, R1_c_list, C.A[0], R2, N, Y)
return S
def low_index_subgroups(G, N, Y=[]):
"""
Implements the Low Index Subgroups algorithm, i.e find all subgroups of
``G`` upto a given index ``N``. This implements the method described in
[Sim94]. This procedure involves a backtrack search over incomplete Coset
Tables, rather than over forced coincidences.
Parameters
==========
G: An FpGroup < X|R >
N: positive integer, representing the maximum index value for subgroups
Y: (an optional argument) specifying a list of subgroup generators, such
that each of the resulting subgroup contains the subgroup generated by Y.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, low_index_subgroups
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**2, y**3, (x*y)**4])
>>> L = low_index_subgroups(f, 4)
>>> for coset_table in L:
... print(coset_table.table)
[[0, 0, 0, 0]]
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]]
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]]
[[1, 1, 0, 0], [0, 0, 1, 1]]
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 5.4
.. [2] Marston Conder and Peter Dobcsanyi
"Applications and Adaptions of the Low Index Subgroups Procedure"
"""
C = CosetTable(G, [])
R = G.relators
# length chosen for the length of the short relators
len_short_rel = 5
# elements of R2 only checked at the last step for complete
# coset tables
R2 = set([rel for rel in R if len(rel) > len_short_rel])
# elements of R1 are used in inner parts of the process to prune
# branches of the search tree,
R1 = set([rel.identity_cyclic_reduction() for rel in set(R) - R2])
R1_c_list = C.conjugates(R1)
S = []
descendant_subgroups(S, C, R1_c_list, C.A[0], R2, N, Y)
return S
def test_modified_methods():
"""
Tests for modified coset table methods.
Example 5.7 from [1] Holt, D., Eick, B., O'Brien
"Handbook of Computational Group Theory".
"""
F, x, y = free_group("x, y")
f = FpGroup(F, [x ** 3, y ** 5, (x * y) ** 2])
H = [x * y, x ** -1 * y ** -1 * x * y * x]
C = CosetTable(f, H)
C.modified_define(0, x)
identity = C._grp.identity
a_0 = C._grp.generators[0]
a_1 = C._grp.generators[1]
assert C.P == [[identity, None, None, None], [None, identity, None, None]]
assert C.table == [[1, None, None, None], [None, 0, None, None]]
C.modified_define(1, x)
assert C.table == [[1, None, None, None], [2, 0, None, None], [None, 1, None, None]]
assert C.P == [
[identity, None, None, None],
[identity, identity, None, None],
[None, identity, None, None],
]
C.modified_scan(0, x ** 3, C._grp.identity, fill=False)
assert C.P == [
[identity, identity, None, None],
[identity, identity, None, None],
[identity, identity, None, None],
]
assert C.table == [[1, 2, None, None], [2, 0, None, None], [0, 1, None, None]]
C.modified_scan(0, x * y, C._grp.generators[0], fill=False)
assert C.P == [
[identity, identity, None, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, None],
]
assert C.table == [[1, 2, None, 1], [2, 0, 0, None], [0, 1, None, None]]
C.modified_define(2, y ** -1)
assert C.table == [
[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[None, None, 2, None],
]
assert C.P == [
[identity, identity, None, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[None, None, identity, None],
]
C.modified_scan(0, x ** -1 * y ** -1 * x * y * x, C._grp.generators[1])
assert C.table == [
[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[3, 3, 2, None],
]
assert C.P == [
[identity, identity, None, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1 ** -1, identity, None],
]
C.modified_scan(2, (x * y) ** 2, C._grp.identity)
assert C.table == [[1, 2, 3, 1], [2, 0, 0, None], [0, 1, None, 3], [3, 3, 2, 0]]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1 ** -1, identity, a_1],
]
C.modified_define(2, y)
assert C.table == [
[1, 2, 3, 1],
[2, 0, 0, None],
[0, 1, 4, 3],
[3, 3, 2, 0],
[None, None, None, 2],
]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, identity, identity],
[a_1, a_1 ** -1, identity, a_1],
[None, None, None, identity],
]
C.modified_scan(0, y ** 5, C._grp.identity)
assert C.table == [
[1, 2, 3, 1],
[2, 0, 0, 4],
[0, 1, 4, 3],
[3, 3, 2, 0],
[None, None, 1, 2],
]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, a_0 * a_1 ** -1],
[identity, identity, identity, identity],
[a_1, a_1 ** -1, identity, a_1],
[None, None, a_1 * a_0 ** -1, identity],
]
C.modified_scan(1, (x * y) ** 2, C._grp.identity)
assert C.table == [
[1, 2, 3, 1],
[2, 0, 0, 4],
[0, 1, 4, 3],
[3, 3, 2, 0],
[4, 4, 1, 2],
]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, a_0 * a_1 ** -1],
[identity, identity, identity, identity],
[a_1, a_1 ** -1, identity, a_1],
[a_0 * a_1 ** -1, a_1 * a_0 ** -1, a_1 * a_0 ** -1, identity],
]
# Modified coset enumeration test
f = FpGroup(F, [x ** 3, y ** 3, x ** -1 * y ** -1 * x * y])
C = coset_enumeration_r(f, [x])
C_m = modified_coset_enumeration_r(f, [x])
assert C_m.table == C.table
def test_scan_1():
# Example 5.1 from [1]
F, x, y = free_group("x, y")
f = FpGroup(F, [x ** 3, y ** 3, x ** -1 * y ** -1 * x * y])
c = CosetTable(f, [x])
c.scan_and_fill(0, x)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, x ** 3)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, y ** 3)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [None, None, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, x ** -1 * y ** -1 * x * y)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [2, 2, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x ** 3)
assert c.table == [
[0, 0, 1, 2],
[3, 4, 2, 0],
[2, 2, 0, 1],
[4, 1, None, None],
[1, 3, None, None],
]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, y ** 3)
assert c.table == [
[0, 0, 1, 2],
[3, 4, 2, 0],
[2, 2, 0, 1],
[4, 1, None, None],
[1, 3, None, None],
]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, x ** -1 * y ** -1 * x * y)
assert c.table == [
[0, 0, 1, 2],
[1, 1, 2, 0],
[2, 2, 0, 1],
[None, 1, None, None],
[1, 3, None, None],
]
assert c.p == [0, 1, 2, 1, 1]
assert c.n == 3
assert c.omega == [0, 1, 2]
# Example 5.2 from [1]
f = FpGroup(F, [x ** 2, y ** 3, (x * y) ** 3])
c = CosetTable(f, [x * y])
c.scan_and_fill(0, x * y)
assert c.table == [[1, None, None, 1], [None, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, x ** 2)
assert c.table == [[1, 1, None, 1], [0, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, y ** 3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, (x * y) ** 3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x ** 2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, y ** 3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, (x * y) ** 3)
assert c.table == [
[1, 1, 2, 1],
[0, 0, 0, 2],
[3, 4, 1, 0],
[None, 2, 4, None],
[2, None, None, 3],
]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(2, x ** 2)
assert c.table == [
[1, 1, 2, 1],
[0, 0, 0, 2],
[3, 3, 1, 0],
[2, 2, 3, 3],
[2, None, None, 3],
]
assert c.p == [0, 1, 2, 3, 3]
assert c.n == 4
assert c.omega == [0, 1, 2, 3]
def test_modified_methods():
'''
Tests for modified coset table methods.
Example 5.7 from [1] Holt, D., Eick, B., O'Brien
"Handbook of Computational Group Theory".
'''
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
C = CosetTable(f, H)
C.modified_define(0, x)
identity = C._grp.identity
a_0 = C._grp.generators[0]
a_1 = C._grp.generators[1]
assert C.P == [[identity, None, None, None],
[None, identity, None, None]]
assert C.table == [[1, None, None, None],
[None, 0, None, None]]
C.modified_define(1, x)
assert C.table == [[1, None, None, None],
[2, 0, None, None],
[None, 1, None, None]]
assert C.P == [[identity, None, None, None],
[identity, identity, None, None],
[None, identity, None, None]]
C.modified_scan(0, x**3, C._grp.identity, fill=False)
assert C.P == [[identity, identity, None, None],
[identity, identity, None, None],
[identity, identity, None, None]]
assert C.table == [[1, 2, None, None],
[2, 0, None, None],
[0, 1, None, None]]
C.modified_scan(0, x*y, C._grp.generators[0], fill=False)
assert C.P == [[identity, identity, None, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, None]]
assert C.table == [[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, None]]
C.modified_define(2, y**-1)
assert C.table == [[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[None, None, 2, None]]
assert C.P == [[identity, identity, None, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[None, None, identity, None]]
C.modified_scan(0, x**-1*y**-1*x*y*x, C._grp.generators[1])
assert C.table == [[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[3, 3, 2, None]]
assert C.P == [[identity, identity, None, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1**-1, identity, None]]
C.modified_scan(2, (x*y)**2, C._grp.identity)
assert C.table == [[1, 2, 3, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[3, 3, 2, 0]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1**-1, identity, a_1]]
C.modified_define(2, y)
assert C.table == [[1, 2, 3, 1],
[2, 0, 0, None],
[0, 1, 4, 3],
[3, 3, 2, 0],
[None, None, None, 2]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, identity, identity],
[a_1, a_1**-1, identity, a_1],
[None, None, None, identity]]
C.modified_scan(0, y**5, C._grp.identity)
assert C.table == [[1, 2, 3, 1], [2, 0, 0, 4], [0, 1, 4, 3], [3, 3, 2, 0], [None, None, 1, 2]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, a_0*a_1**-1],
[identity, identity, identity, identity],
[a_1, a_1**-1, identity, a_1],
[None, None, a_1*a_0**-1, identity]]
C.modified_scan(1, (x*y)**2, C._grp.identity)
assert C.table == [[1, 2, 3, 1],
[2, 0, 0, 4],
[0, 1, 4, 3],
[3, 3, 2, 0],
[4, 4, 1, 2]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, a_0*a_1**-1],
[identity, identity, identity, identity],
[a_1, a_1**-1, identity, a_1],
[a_0*a_1**-1, a_1*a_0**-1, a_1*a_0**-1, identity]]
# Modified coset enumeration test
f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
C = coset_enumeration_r(f, [x])
C_m = modified_coset_enumeration_r(f, [x])
assert C_m.table == C.table
def test_scan_1():
# Example 5.1 from [1]
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
c = CosetTable(f, [x])
c.scan_and_fill(0, x)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, x**3)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, y**3)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [None, None, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, x**-1*y**-1*x*y)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [2, 2, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x**3)
assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \
[4, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, y**3)
assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \
[4, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, x**-1*y**-1*x*y)
assert c.table == [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], \
[None, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 1, 1]
assert c.n == 3
assert c.omega == [0, 1, 2]
# Example 5.2 from [1]
f = FpGroup(F, [x**2, y**3, (x*y)**3])
c = CosetTable(f, [x*y])
c.scan_and_fill(0, x*y)
assert c.table == [[1, None, None, 1], [None, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, x**2)
assert c.table == [[1, 1, None, 1], [0, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, y**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, (x*y)**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x**2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, y**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, (x*y)**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 4, 1, 0], [None, 2, 4, None], [2, None, None, 3]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(2, x**2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 3, 1, 0], [2, 2, 3, 3], [2, None, None, 3]]
assert c.p == [0, 1, 2, 3, 3]
assert c.n == 4
assert c.omega == [0, 1, 2, 3]