File size: 10,203 Bytes
2902979 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 |
from sympy.core.singleton import S
from sympy.combinatorics.fp_groups import (FpGroup, low_index_subgroups,
reidemeister_presentation, FpSubgroup,
simplify_presentation)
from sympy.combinatorics.free_groups import (free_group, FreeGroup)
from sympy.testing.pytest import slow
"""
References
==========
[1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
[2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
"Implementation and Analysis of the Todd-Coxeter Algorithm"
[3] PROC. SECOND INTERNAT. CONF. THEORY OF GROUPS, CANBERRA 1973,
pp. 347-356. "A Reidemeister-Schreier program" by George Havas.
http://staff.itee.uq.edu.au/havas/1973cdhw.pdf
"""
def test_low_index_subgroups():
F, x, y = free_group("x, y")
# Example 5.10 from [1] Pg. 194
f = FpGroup(F, [x**2, y**3, (x*y)**4])
L = low_index_subgroups(f, 4)
t1 = [[[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]]]
for i in range(len(t1)):
assert L[i].table == t1[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 15)
t2 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[4, 4, 5, 3], [6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [9, 9, 6, 8], [12, 12, 11, 7], [8, 8, 7, 10],
[10, 10, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [10, 10, 11, 7], [8, 8, 7, 10],
[9, 9, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [13, 13, 11, 7], [8, 8, 7, 10],
[9, 9, 12, 12], [10, 10, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [11, 11, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 4, 4]
, [7, 7, 6, 3], [8, 8, 3, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [5, 5, 6, 3], [9, 9, 3, 5], [10, 10, 8, 4], [8, 8, 4, 7],
[6, 6, 10, 11], [7, 7, 11, 9], [12, 12, 9, 10], [11, 11, 13, 14],
[14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 10, 12], [7, 7, 12, 9], [8, 8, 11, 11], [13, 13, 9, 10],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [7, 7, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 9, 9], [6, 6, 11, 12], [8, 8, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [6, 6, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [11, 11, 8, 4], [12, 12, 4, 7],
[5, 5, 9, 9], [6, 6, 12, 13], [7, 7, 11, 11], [8, 8, 13, 10],
[13, 13, 10, 12]],
[[1, 1, 0, 0], [0, 0, 2, 3], [4, 4, 3, 1], [5, 5, 1, 2], [2, 2, 4, 4]
, [3, 3, 6, 7], [7, 7, 7, 5], [6, 6, 5, 6]]]
for i in range(len(t2)):
assert L[i].table == t2[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 10, [x])
t3 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [4, 4, 5, 3],
[6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [6, 6, 5, 3],
[5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8],
[6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]]]
for i in range(len(t3)):
assert L[i].table == t3[i]
def test_subgroup_presentations():
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]
p1 = reidemeister_presentation(f, H)
assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"
H = f.subgroup(H)
assert (H.generators, H.relators) == p1
f = FpGroup(F, [x**3, y**3, (x*y)**3])
H = [x*y, x*y**-1]
p2 = reidemeister_presentation(f, H)
assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"
f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
H = [x]
p3 = reidemeister_presentation(f, H)
assert str(p3) == "((x_0,), (x_0**4,))"
f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
H = [x]
p4 = reidemeister_presentation(f, H)
assert str(p4) == "((x_0,), (x_0**6,))"
# this presentation can be improved, the most simplified form
# of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
# See [2] Pg 474 group PSL_2(11)
# This is the group PSL_2(11)
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
H = [a, b, c**2]
gens, rels = reidemeister_presentation(f, H)
assert str(gens) == "(b_1, c_3)"
assert len(rels) == 18
@slow
def test_order():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.order() == 8
f = FpGroup(F, [x*y*x**-1*y**-1, y**2])
assert f.order() is S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**250, b**2, c*b*c**-1*b, c**4, c**-1*a**-1*c*a, a**-1*b**-1*a*b])
assert f.order() == 2000
F, x = free_group("x")
f = FpGroup(F, [])
assert f.order() is S.Infinity
f = FpGroup(free_group('')[0], [])
assert f.order() == 1
def test_fp_subgroup():
def _test_subgroup(K, T, S):
_gens = T(K.generators)
assert all(elem in S for elem in _gens)
assert T.is_injective()
assert T.image().order() == S.order()
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
S = FpSubgroup(f, [x*y])
assert (x*y)**-3 in S
K, T = f.subgroup([x*y], homomorphism=True)
assert T(K.generators) == [y*x**-1]
_test_subgroup(K, T, S)
S = FpSubgroup(f, [x**-1*y*x])
assert x**-1*y**4*x in S
assert x**-1*y**4*x**2 not in S
K, T = f.subgroup([x**-1*y*x], homomorphism=True)
assert T(K.generators[0]**3) == y**3
_test_subgroup(K, T, S)
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
K, T = f.subgroup(H, homomorphism=True)
S = FpSubgroup(f, H)
_test_subgroup(K, T, S)
def test_permutation_methods():
F, x, y = free_group("x, y")
# DihedralGroup(8)
G = FpGroup(F, [x**2, y**8, x*y*x**-1*y])
T = G._to_perm_group()[1]
assert T.is_isomorphism()
assert G.center() == [y**4]
# DiheadralGroup(4)
G = FpGroup(F, [x**2, y**4, x*y*x**-1*y])
S = FpSubgroup(G, G.normal_closure([x]))
assert x in S
assert y**-1*x*y in S
# Z_5xZ_4
G = FpGroup(F, [x*y*x**-1*y**-1, y**5, x**4])
assert G.is_abelian
assert G.is_solvable
# AlternatingGroup(5)
G = FpGroup(F, [x**3, y**2, (x*y)**5])
assert not G.is_solvable
# AlternatingGroup(4)
G = FpGroup(F, [x**3, y**2, (x*y)**3])
assert len(G.derived_series()) == 3
S = FpSubgroup(G, G.derived_subgroup())
assert S.order() == 4
def test_simplify_presentation():
# ref #16083
G = simplify_presentation(FpGroup(FreeGroup([]), []))
assert not G.generators
assert not G.relators
# CyclicGroup(3)
# The second generator in <x, y | x^2, x^5, y^3> is trivial due to relators {x^2, x^5}
F, x, y = free_group("x, y")
G = simplify_presentation(FpGroup(F, [x**2, x**5, y**3]))
assert x in G.relators
def test_cyclic():
F, x, y = free_group("x, y")
f = FpGroup(F, [x*y, x**-1*y**-1*x*y*x])
assert f.is_cyclic
f = FpGroup(F, [x*y, x*y**-1])
assert f.is_cyclic
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert not f.is_cyclic
def test_abelian_invariants():
F, x, y = free_group("x, y")
f = FpGroup(F, [x*y, x**-1*y**-1*x*y*x])
assert f.abelian_invariants() == []
f = FpGroup(F, [x*y, x*y**-1])
assert f.abelian_invariants() == [2]
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.abelian_invariants() == [2, 4]
|