def is_subgroup(self, other):
r"""
Return True if self is a subgroup of right, and False
otherwise.
EXAMPLES::
sage: GammaH(24,[7]).is_subgroup(SL2Z)
True
sage: GammaH(24,[7]).is_subgroup(Gamma0(8))
True
sage: GammaH(24, []).is_subgroup(GammaH(24, [7]))
True
sage: GammaH(24, []).is_subgroup(Gamma1(24))
True
sage: GammaH(24, [17]).is_subgroup(GammaH(24, [7]))
False
sage: GammaH(1371, [169]).is_subgroup(GammaH(457, [169]))
True
"""
from all import is_Gamma0, is_Gamma1
if not isinstance(other, GammaH_class):
raise NotImplementedError
# level of self should divide level of other
if self.level() % other.level():
return False
# easy cases
if is_Gamma0(other):
return True # recall self is a GammaH, so it's contained in Gamma0
if is_Gamma1(other) and len(self._generators_for_H()) > 0:
return False
else:
# difficult case
t = other._list_of_elements_in_H()
for x in self._generators_for_H():
if not (x in t):
return False
return True
def is_subgroup(self, other):
r"""
Return True if self is a subgroup of right, and False
otherwise.
EXAMPLES::
sage: GammaH(24,[7]).is_subgroup(SL2Z)
True
sage: GammaH(24,[7]).is_subgroup(Gamma0(8))
True
sage: GammaH(24, []).is_subgroup(GammaH(24, [7]))
True
sage: GammaH(24, []).is_subgroup(Gamma1(24))
True
sage: GammaH(24, [17]).is_subgroup(GammaH(24, [7]))
False
sage: GammaH(1371, [169]).is_subgroup(GammaH(457, [169]))
True
"""
from all import is_Gamma0, is_Gamma1
if not isinstance(other, GammaH_class):
raise NotImplementedError
# level of self should divide level of other
if self.level() % other.level():
return False
# easy cases
if is_Gamma0(other):
return True # recall self is a GammaH, so it's contained in Gamma0
if is_Gamma1(other) and len(self._generators_for_H()) > 0:
return False
else:
# difficult case
t = other._list_of_elements_in_H()
for x in self._generators_for_H():
if not (x in t):
return False
return True
def __cmp__(self, other):
"""
Compare self to other.
The ordering on congruence subgroups of the form GammaH(N) for
some H is first by level and then by the subgroup H. In
particular, this means that we have Gamma1(N) < GammaH(N) <
Gamma0(N) for every nontrivial subgroup H.
EXAMPLES::
sage: G = Gamma0(86)
sage: G.__cmp__(G)
0
sage: G.__cmp__(GammaH(86, [11])) is not 0
True
sage: Gamma1(17) < Gamma0(17)
True
sage: Gamma0(1) == SL2Z
True
sage: Gamma0(11) == GammaH(11, [2])
True
sage: Gamma0(2) == Gamma1(2)
True
"""
if not is_CongruenceSubgroup(other):
return cmp(type(self), type(other))
c = cmp(self.level(), other.level())
if c: return c
# Since Gamma0(N) is GammaH(N) for H all of (Z/N)^\times,
# we know how to compare it to any other GammaH without having
# to look at self._list_of_elements_in_H().
from all import is_GammaH, is_Gamma0
if is_GammaH(other):
if is_Gamma0(other):
return 0
else:
H = other._list_of_elements_in_H()
return cmp(len(H), arith.euler_phi(self.level()))
return cmp(type(self), type(other))
def is_subgroup(self, right):
"""
Return True if self is a subgroup of right.
EXAMPLES::
sage: G = Gamma0(20)
sage: G.is_subgroup(SL2Z)
True
sage: G.is_subgroup(Gamma0(4))
True
sage: G.is_subgroup(Gamma0(20))
True
sage: G.is_subgroup(Gamma0(7))
False
sage: G.is_subgroup(Gamma1(20))
False
sage: G.is_subgroup(GammaH(40, []))
False
sage: Gamma0(80).is_subgroup(GammaH(40, [31, 21, 17]))
True
sage: Gamma0(2).is_subgroup(Gamma1(2))
True
"""
if right.level() == 1:
return True
if is_Gamma0(right):
return self.level() % right.level() == 0
if is_Gamma1(right):
if right.level() >= 3:
return False
elif right.level() == 2:
return self.level() == 2
# case level 1 dealt with above
else:
return GammaH_class.is_subgroup(self, right)
