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[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.