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# 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

• Set

(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

• 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.