[index] Algebra::OperatorDomain / Algebra::Set / Algebra::Group / Algebra::QuotientGroup

Algebra::OperatorDomain

This is the module for the set oprated by groups. This is included by Group.

File Name:

Methods:

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 *.

Algebra::Set

File Name:

Included Module:

Methods:

* 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

Algebra::Group

File Name:

SuperClass:

Included Module:

(None)

Class Methods:

::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.

Methods:

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.

Algebra::QuotientGroup

File Name:

SuperClass:

Class Methods:

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.

Methods:

inverse
Returns the inverse element.
inv
Alias of inverse.