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

Algebra::OperatorDomain

Ѥ뽸򽸤᤿⥸塼Ǥ Group 饹󥯥롼ɤƤ
ޤ

ե̾:

  finite-group.rb

᥽å:

right_act(other)
    self  other Ѥ֤ޤʤ self θ x  other θ y Ф
    x * y ηθνSetˤ֤ޤ
   
act
    right_act ΥꥢǤ
   
left_act(other)
    self  other Ѥ֤ޤʤ self θ x  other θ y Ф
    y * x ηθνSetˤ֤ޤ
   
right_quotient(other)
    self  other ǳä;νSetˤ֤ޤ
   
quotient
right_coset
coset
    right_quotient ΥꥢǤ
   
left_quotient(other)
    self  other ǳä;νSetˤ֤ޤ
   
left_coset
    left_quotient ΥꥢǤ
   
right_representatives(other)
    ; right_quotient äɽν֤ޤ
   
representatives
    right_representatives ΥꥢǤ
   
left_representatives(other)
    ; left_quotient äɽν֤ޤ
   
right_orbit!(other)
    self  other 򷫤֤ѤƹޤѤ * ˤޤ
   
orbit!
    right_orbit! ΥꥢǤ
   
left_orbit!(other)
    self  other 򷫤֤ѤƹޤѤ * ˤޤ
   

Algebra::Set

ե̾:

  finite-group.rb Ǥ finite-set.rb 줿 Set դä٤
    ᥽åɤƤޤ

󥯥롼ɤƤ⥸塼:

  OperatorDomain

᥽å:

*
    act ΥꥢǤ
   
/
    quotient ΥꥢǤ
   
%
    representatives ΥꥢǤ
   
increasing_series([x])
    x Ϥޤ֤ޤϼΥɤƱͤǤ
    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])
    x Ϥޤ븺֤ޤϼΥɤƱͤǤ
    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

ե̾:

  finite-group.rb

ѡ饹:

  Set

饹᥽å:

::new(u, [g0, g1, ...]])
    u ñ̸Ȥg0, g1, ... ǹ뷲֤ޤ
   
::generate_strong(u, [g0, [g1, ...]])
    ñ̸ u g0, g1, ... Ȥơ뷲֤ޤ
   

᥽å:

quotient_group(u)
    ʬ u ˤ;֤ޤ
   
separate
    ֥å򿿤Ȥ븵ʤʬ֤ޤ
   
to_a
    Ǥˤ֤ޤǽǤñ̸Ǥ
   
unity
    ñ̸֤ޤ
   
unit_group
    ñ̸ñ̷֤ޤ
   
semi_complete!
    ߤǤݤ碌Ⱦޤ
   
semi_complete
    ߤǤݤ碌ȾΤ֤ޤ
   
complete!
    ߤǤݤ碌Ʒޤ
   
complete
    ߤǤݤ碌ƷΤ֤ޤ
   
closed?
    ȤĤƤȤ֤ޤ
   
subgroups
    Ƥʬν֤ޤ
   
centralizer(s)
    self ˤ s 濴֤ޤ
   
center
    self 濴֤ޤ
   
center?(x)
    self Ǹ x 濴˴ޤޤƤȤ֤ޤ
   
normalizer(s)
    self ˤ s ֤ޤ
   
normal?(s)
    s  self ʬǤȤ֤ޤ
   
normal_subgroups
    Ƥʬν֤ޤ
   
conjugacy_class(x)
     x ζ֤ޤ
   
conjugacy_classes
    self Ƥζν֤ޤ
   
simple?
    ñ㷲ǤȤ֤ޤ
   
commutator([h])
    self  h Ȥθ򴹻ҷ֤ޤh ά self Ѥޤ
   
D([n])
    n ܤθ򴹻ҷ֤ޤD(0) = ʬ, D(n+1) = [D[n], D[n]] 
    Ƥޤ n ά 1 Ѥޤ
   
commutator_series
    [D(0), D(1), D(2),..., D(n)] Ȥ֤ޤ D(n) == D
    (n+1) Ȥʤ n ߤޤ
   
solvable?
    Ĳ򷲤ǤȤ֤ޤ
   
K([n])
    K(0) = ʬ, K(n+1) = [self, K[n]] 뷲֤ޤ n ά
     1 Ѥޤ
   
descending_central_series
    濴 [K(0), K(1), K(2),..., K(n)] Ȥ֤ޤ K(n)
    == K(n+1) Ȥʤ n ߤޤ
   
Z([n])
    Z(0) = ñ̷, Z(n+1) = separate{|x| commutator(Set[x]) <= Z(n-1)} 
    뷲֤ޤ n ά 1 Ѥޤ
   
ascending_central_series
    濴 [Z(0), Z(1), Z(2),..., Z(n)] Ȥ֤ޤ Z(n)
    == Z(n+1) Ȥʤ n ߤޤ
   
nilpotent?
    ǤȤ֤ޤ
   
nilpotency_class
    饹֤ޤ㷲ǤʤȤ nil ֤ޤ
   

Algebra::QuotientGroup

ե̾:

  finite-group.rb

ѡ饹:

  Group

饹᥽å:

new(u, [g0, [g1,...]])
    self ʬ u Ȥơ u, g0, g1, .. ;Ȥ;֤ޤ
    
   

᥽å:

inverse
    ո֤ޤ
   
inv
    inverse ΥꥢǤ
   

