[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.
actAlias 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
cosetAlias of right_quotient.
left_quotient(other)Returns the Set of left residue classes of self by other.
left_cosetAlias of left_quotient.
right_representatives(other)Returns the representatives of the right residue classes right_quotient.
representativesAlias 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 *.
* actAlias 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
enddecreasing_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.
separateReturns the subgroup whose elements makes the block true.
to_aReturns the array of elements. The first is the unity.
unityReturns the unity.
unit_groupReturns the unit group.
semi_complete!Makes self be the semi-group generated by the elements.
semi_completeReturns the semi-group generated by the elements.
complete!Makes self be the semi-group generated by the elements.
completeReturns the group generated by the elements.
closed?Returns true when self is closed by product and inverse.
subgroupsReturns the all subgroups.
centralizer(s)Returns the centralize of s in self.
centerReturns 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_subgroupsReturns the all normal subgroups.
conjugacy_class(x)Returns the conjugacy class of the element x.
conjugacy_classesReturns 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_seriesReturns 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_seriesReturns 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_seriesReturns 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_classReturns 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.
inverseReturns the inverse element.
invAlias of inverse.