def Dih(n):
'Dihedral group of order n'
if n == 1:
return simple.z(2)
elif n == 2:
c = numpy.array([[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 0, 1],
[3, 2, 1, 0]])
return algebra.algebra(c)
elif n == 3:
c = numpy.array([[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3],
[2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1],
[4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]])
return algebra.algebra(c)
else:
raise Exception("Dihedral groups so far only defined for n<4")
def randomGroup(n):
'''Randomly generate a group of order n. It does that by putting in random
values and then checking group axioms. This is about as inefficient as an
algorithm can possibly get!!!
'''
count = 0 # We need some exit mechanism in case it's impossible
while True:
c = numpy.empty((n, n), dtype=numpy.uint32)
for i, j in itertools.product(range(n), range(n)):
if i == 0:
c[i, j] = j
elif j == 0:
c[i, j] = i
else:
c[i, j] = random.randint(0, n-1)
is_good = True
# Check invertibility
for i in range(n):
has_inverse = False
for j in range(n):
if c[i, j] == 0 and c[j, i] == 0:
has_inverse = True
break
if not has_inverse:
is_good = False
break
# Check associativity
if is_good:
for i, j, k in itertools.product(range(n), range(n), range(n)):
if c[c[i, j], k] != c[i, c[j, k]]:
is_good = False
break
if is_good:
return algebra(c)
# If not good, go around and try another one...
count += 1
if count > 1000000:
raise Exception("Didn't find a group of order {0}!".format(n))
def Steinberg(family, n, q):
order = 0
if family == '2A':
# Classic Steinberg group
if n <= 1:
raise Exception
x = 1
for i in range(1, n + 1):
x *= (q**(i + 1) - (-1)**(i + 1))
x *= q**(n * (n + 1) // 2)
x /= math.gcd(n + 1, q + 1)
name = f"^(2)A_{n}({q}^2)"
order = x
elif family == '2D':
# Classic Steinberg group
if n <= 3:
raise Exception
x = 1
for i in range(1, n + 1):
x *= (q**(2 * i) - 1)
x *= (q**2 + 1)
x *= q**(n * (n - 1))
x /= math.gcd(4, q**n + 1)
name = f"^(2)D_{n}({q}^2)"
order = x
elif family == '2E':
# Exceptional Steinberg group
if n != 6:
raise Exception
x = 1
for i in [2, 5, 6, 8, 9, 12]:
x *= (q**i - (-1)**i)
x *= q**36
x /= math.gcd(3, q + 1)
name = f"^(2)E_{n}({q}^2)"
order = x
elif family == '3E':
# Exceptional Steinberg group
if n != 4:
raise Exception
x = 1
x *= q**12
x *= (q**8 + q**4 + 1)
x *= (q**6 - 1)
x *= (q**2 - 1)
name = f"^(3)E_{n}({q}^3)"
order = x
c = algebra(None)
c.setName(name)
c.setOrder(order)
return c
def Suzuki(q):
n = 0.5 * (math.log(q, 2) - 1)
if abs(
n - round(n)
) > 1.E-7: # This "epsilon" value is only suitable for quite small values of q. Revisit...
raise Exception("q must be of form 2^(2n+1). q={0}, n={1}".format(
q, n))
name = f"^(2)B_2({q})"
order = q**2 * (q**2 + 1) * (q - 1)
c = algebra(None)
c.setName(name)
c.setOrder(order)
return c
def z(n):
'''
Cyclic group of order n.
'''
if n <= 0:
raise Exception("Need a positive order")
c = numpy.empty((n, n), dtype=numpy.uint32)
for i in range(n):
x = i
for j in range(n):
c[i, j] = x
x += 1
if x >= n:
x -= n
return algebra(c)
def Chevalley(family, n, q):
order = 0
if family == 'A':
# Classic Chevalley group
x = 1
for i in range(1, n + 1):
x *= (q**(i + 1) - 1)
x *= q**(n * (n + 1) // 2)
x /= math.gcd(n + 1, q - 1)
name = f"A_{n}({q})"
order = x
elif family == 'B':
# Classic Chevalley group
if n < 1:
raise Exception
x = 1
for i in range(1, n + 1):
x *= (q**(2 * i) - 1)
x *= q**(n * n)
x /= math.gcd(2, q - 1)
name = f"B_{n}({q})"
order = x
elif family == 'C':
# Classic Chevalley group
if n < 2:
raise Exception
x = 1
for i in range(1, n + 1):
x *= (q**(2 * i) - 1)
x *= q**(n * n)
x /= math.gcd(2, q - 1)
name = f"C_{n}({q})"
order = x
elif family == 'D':
# Classic Chevalley group
if n < 2:
raise Exception
x = 1
for i in range(1, n + 1):
x *= (q**(2 * i) - 1)
x *= q**(n * (n - 1)) * (q**n - 1)
x /= math.gcd(4, q**n - 1)
name = f"D_{n}({q})"
order = x
elif family == 'E':
# Exceptional Chevalley group
x = 1
if n == 6:
for i in [2, 5, 6, 8, 9, 12]:
x *= (q**i - 1)
x *= q**36
x /= math.gcd(3, q - 1)
elif n == 7:
for i in [2, 6, 8, 10, 12, 14, 18]:
x *= (q**i - 1)
x *= q**63
x /= math.gcd(2, q - 1)
elif n == 8:
for i in [2, 8, 12, 14, 18, 20, 24, 30]:
x *= (q**i - 1)
x *= q**120
else:
raise Exception("Impossible value for n")
name = f"E_{n}({q})"
order = x
elif family == 'F':
# Exceptional Chevalley group
x = 1
if n == 4:
for i in [2, 6, 8, 12]:
x *= (q**i - 1)
x *= q**24
else:
raise Exception("Impossible value for n")
name = f"F_{n}({q})"
order = x
elif family == 'G':
# Exceptional Chevalley group
x = 1
if n == 2:
for i in [2, 6]:
x *= (q**i - 1)
x *= q**6
else:
raise Exception("Impossible value for n")
name = f"G_{n}({q})"
order = x
c = algebra(None)
c.setName(name)
c.setOrder(order)
return c