[index]
Algebra::OperatorDomain
/
Algebra::Set
/
Algebra::Group
/
Algebra::QuotientGroup
This is the module for the set oprated by groups.
This is included by Group.
right_act(other)
-
Returns the products of self and other, i.e.
Set of
x * y
for
x element of self and y element of other.
act
-
Alias of right_act.
left_act(other)
-
Returns the products of self and other, i.e.
Set of
y * x
for
x element of self and y element of other.
right_quotient(other)
-
Returns the Set of right residue classes
of self by other.
quotient
right_coset
coset
-
Alias of right_quotient.
left_quotient(other)
-
Returns the Set of left residue classes
of self by other.
left_coset
-
Alias of left_quotient.
right_representatives(other)
-
Returns the representatives of the
right residue classes right_quotient.
representatives
-
Alias of right_representatives.
left_representatives(other)
-
Returns the representatives of the
left residue classes left_quotient.
right_orbit!(other)
-
Extends self operating the elements of other by right action
*.
orbit!
-
Alias of right_orbit!.
left_orbit!(other)
-
Extends self operating the elements of other by left action
*.
* act
-
Alias of act
/
-
Alias of quotient.
%
-
Alias of representatives.
increasing_series([x])
-
Returns the increasing series begining with x.
This is equivalent to the following code:
def increasing_series(x = unit_group)
a = []
loop do
a.push x
if x >= (y = yield x)
break
end
x = y
end
a
end
decreasing_series([x])
-
Returns the decreasing series begining with x.
This is equivalent to the following code:
def decreasing_series(x = self)
a = []
loop do
a.push x
if x <= (y = yield x)
break
end
x = y
end
a
end
(None)
::new(u, [g0, g1, ...]])
-
Returns the group which consists of u, g0, g1, ...
and whose unity is u.
::generate_strong(u, [g0, [g1, ...]])
-
Returns the group strongly generated by g0, g1, ...
and whose unity is u.
quotient_group(u)
-
Returns the residue class group of the normal subgroup u.
separate
-
Returns the subgroup whose elements makes the block true.
to_a
-
Returns the array of elements. The first is the unity.
unity
-
Returns the unity.
unit_group
-
Returns the unit group.
semi_complete!
-
Makes self be the semi-group generated by the elements.
semi_complete
-
Returns the semi-group generated by the elements.
complete!
-
Makes self be the semi-group generated by the elements.
complete
-
Returns the group generated by the elements.
closed?
-
Returns true when self is closed by product and inverse.
subgroups
-
Returns the all subgroups.
centralizer(s)
-
Returns the centralize of s in self.
center
-
Returns the center ofself.
center?(x)
-
Returns true if x is in the center of self.
normalizer(s)
-
Returns the normalizer of s in self.
normal?(s)
-
Returns true if s is a normal subgroup of self.
normal_subgroups
-
Returns the all normal subgroups.
conjugacy_class(x)
-
Returns the conjugacy class of the element x.
conjugacy_classes
-
Returns the set of all conjucacy claases of self.
simple?
-
Retuns true if self is a simple group.
commutator([h])
-
Returns the commutator subgroup of self and h.
If the parameter is omitted, h is assumed to be self.
D([n])
-
Returns the n-the commutator subgroup.
D(0) = self
and D(n+1) = [D[n], D[n]]
.
If the parameter ommitted, n is assumed to be 1.
commutator_series
-
Returns the array
[D(0), D(1), D(2),..., D(n)]
.
This sequence is terminated for n with D(n) == D(n+1)
.
solvable?
-
Returns true if self is solvable.
K([n])
-
Returns the subgroup definend such that
K(0) = self
and
K(n+1) = [self, K[n]
.
If the parameter is omitted, n is asumed to be 1.
descending_central_series
-
Returns the descending central series:
[K(0), K(1), K(2),..., K(n)]
.
This sequence is terminated for n with K(n) == K(n+1)
.
Z([n])
-
Returns the subgroup that defined by:
Z(0) = unit group
,
Z(n+1) = separate{|x| commutator(Set[x]) <= Z(n-1)}
.
If the parameter is omitted, n is assumed to be 1.
ascending_central_series
-
Returns the array of ascending central series:
[Z(0), Z(1), Z(2),..., Z(n)]
.
This sequence is terminated for n such that
Z(n) == Z(n+1)
.
nilpotent?
-
Returns true if self is nilpotent.
nilpotency_class
-
Returns the class of nilpotency.
If self is not nilpotent, returns nil.
new(u, [g0, [g1,...]])
-
Returns the residue class group by u of which the residues are
u, g0, g1, ... Here u is assumed to be
the normal subgroup of self.
inverse
-
Returns the inverse element.
inv
-
Alias of inverse.