def make_random_homogeneous():
# use rotation group
#seed(1)
ngens = 6
a, ai, b, bi, c, ci = range(ngens)
i_rels = [
(ai, a),
(bi, b),
(ci, c),
(a, ) * 5,
(b, ) * 2,
(c, ) * 4,
(a, b, c),
]
while 1:
rels = []
for i in range(argv.get("nwords", 2)):
gen = tuple(
choice([a, b, c]) for k in range(argv.get("wordlen", 100)))
rels.append(gen)
gen = (ci, ) + gen + (c, )
rels.append(gen)
gen = (ci, ) + gen + (c, )
rels.append(gen)
rels = [reduce_word(i_rels, rel) for rel in rels]
graph = Schreier(ngens, i_rels)
if not graph.build(rels, maxsize=10400):
print(choice("/\\"), end="", flush=True)
continue
n = len(graph)
if n <= argv.get("minsize", 12):
print('.', end="", flush=True)
continue
print("[%d]" % n, flush=True, end="")
# how big did the graph get? compare with maxsize above
print("(%d) " % len(graph.neighbors), flush=True, end="")
try:
gens = graph.get_gens()
except AssertionError:
print("** FAIL **")
continue
items = list(range(n))
print()
break
N = argv.get("N", 10000)
perms = mulclose(gens, maxsize=N)
if len(perms) >= N:
#print("|G| = ?")
continue
print()
print(rels)
G = Group(perms, items)
G.gens = gens
print("|G| =", len(G))
rels = [reduce(mul, [gens[i] for i in rel]) for rel in rels]
H = Coset(mulclose(rels), items)
print("|H| =", len(H))
assert len(G) % len(H) == 0
print("[H:G] = ", len(G) // len(H))
N = []
for g in G:
lhs = g * H
rhs = H * g
if lhs == rhs:
N.append(g)
assert len(N) % len(H) == 0
print("[H:N(H)] =", len(N) // len(H))
print(rels)
a, ai, b, bi, c, ci = gens
assert (a**5).is_identity()
assert (b**2).is_identity()
assert (c**4).is_identity()
assert (a * b * c).is_identity()
print(a.fixed())
print(b.fixed())
print(c.fixed())
print(a.orbits())
print(b.orbits())
print(c.orbits())
def make_random_homogeneous_refl():
seed(1)
ngens = 6
a, ai, b, bi, c, ci = range(ngens)
i_rels = [
(ai, a),
(bi, b),
(ci, c),
(a, ) * 2,
(b, ) * 2,
(c, ) * 2,
(a, b) * 5,
(b, c) * 5,
(a, c) * 2,
]
while 1:
rels = []
for i in range(4):
gen = ()
for k in range(20):
gen += choice([(a, b), (a, c), (b, c)])
rels.append(gen)
# G/H = 720/24 = 30, N(H)=24
_rels = [(0, 2, 2, 4, 0, 4, 2, 4, 2, 4, 0, 4, 0, 4, 2, 4, 0, 2, 0, 4,
2, 4, 0, 4, 2, 4, 0, 2, 0, 2, 0, 2, 2, 4, 2, 4, 2, 4, 2, 4),
(0, 2, 2, 4, 0, 4, 2, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2,
0, 2, 0, 2, 0, 4, 0, 4, 2, 4, 0, 4, 0, 2, 0, 2, 2, 4, 0, 2),
(0, 2, 2, 4, 2, 4, 2, 4, 2, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4,
0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 2, 4, 0, 4, 0, 2, 0, 4, 0, 2),
(0, 2, 0, 4, 0, 4, 2, 4, 0, 4, 0, 2, 0, 2, 2, 4, 0, 4, 0, 4,
0, 2, 0, 4, 0, 2, 0, 2, 2, 4, 0, 2, 2, 4, 2, 4, 0, 4, 0, 2)]
# G/H = 9720/324 = 30, N(H)=1944
_rels = [(0, 4, 0, 2, 0, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 0, 4, 0, 2,
0, 2, 0, 4, 2, 4, 0, 2, 0, 2, 0, 2, 2, 4, 0, 2, 0, 4, 0, 4),
(0, 4, 2, 4, 0, 2, 0, 2, 0, 2, 2, 4, 0, 2, 0, 2, 2, 4, 2, 4,
0, 4, 2, 4, 0, 4, 2, 4, 0, 4, 0, 2, 2, 4, 0, 2, 2, 4, 0, 2),
(0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 2, 4, 0, 4, 0, 2, 0, 2,
0, 4, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 4, 2, 4, 0, 2, 2, 4),
(2, 4, 2, 4, 0, 4, 2, 4, 0, 2, 0, 4, 0, 4, 2, 4, 2, 4, 2, 4,
2, 4, 2, 4, 0, 4, 0, 4, 2, 4, 2, 4, 2, 4, 0, 4, 0, 2, 2, 4)]
graph = Schreier(ngens, i_rels)
if not graph.build(rels, maxsize=10400):
print(choice("/\\"), end="", flush=True)
continue
n = len(graph)
if n <= 24:
continue
print("\n[%d]" % n) #, flush=True, end="")
print(len(graph.neighbors)
) # how big did the graph get? compare with maxsize above
try:
gens = graph.get_gens()
except AssertionError:
print("** FAIL **")
continue
N = 10000
perms = mulclose(gens, maxsize=N)
if len(perms) >= N:
print("|G| = ?")
continue
print(rels)
items = list(range(n))
G = Group(perms, items)
G.gens = gens
print("|G| =", len(G))
print()
break
rels = [reduce(mul, [gens[i] for i in rel]) for rel in rels]
H = Coset(mulclose(rels), items)
print(len(H))
print(len(G) / len(H))
N = []
for g in G:
lhs = g * H
rhs = H * g
if lhs == rhs:
N.append(g)
print(len(N))