
[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:

 ¡¦ finite-group.rb

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:

 ¡¦ finite-group.rb

Included Module:

 ¡¦ OperatorDomain

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:

 ¡¦ finite-group.rb

SuperClass:

 ¡¦ Set

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:

 ¡¦ finite-group.rb

SuperClass:

 ¡¦ Group

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.
   

