[index] Algebra::PermutationGroup / Algebra::Permutation
This is the class of permutations. The elements are assumed to be the instances of Permutation.
::new(u, [g0, [g1, ...]])
Returns the group with unit u, whcih consists of g0, g1, ....
::unit_group(d)
Return the unit group of degree d.
::unity(n)
Retunrs the unity of degree n.
::perm(a)
Returns the permuation represented by the array a.
::symmetric(n)
Returns the simmetric group of degree n
::alternate(n)
Returns the alternative group of dgree n.
::new(x)
Returns the permutaiont represented by the array x.
::[[n0, [n1, [n2, ..., ]]]]
Returns the permutation [n0, n1, n2, ..., ]
.
Example:
a = Permutation[1, 2, 0] p a**2 #=> [2, 0, 1] p a**3 #=> [0, 1, 2]
::unity(d)
Returns the unity of degree d.
::cyclic2perm(c, n)
Returns the Permutation represented by c : the array of arrays of cyclic permutations, where n is the degree. This method is the inverse of decompose_cyclic.
Example:
Permutation.cyclic2perm([[1,6,5,4], [2,3]], 7) #=> [0, 6, 3, 2, 1, 4, 5] Permutation[0, 6, 3, 2, 1, 4, 5].decompose_cyclic #=> [[1,6,5,4], [2,3]]
unity
Returns the unity.
perm
Returns the array which represents self
degree
Returns the degree
size
Alias of degree.
each
Iterates for each entry.
eql?(other)
Returns true if self is equal to other.
==
Alias of eql?.
hash
Returns the hash number.
[i]
Returns the number to which i is transferrd.
call
Alias of [].
index(i)
Returns the number from which i is transferred.
right_act(other)
Returns the value multiplied by other from right.
It follows (g.right_act(h))[x] == h[g[x]]
.
*
Alias of right_act
left_act(other)
Returns the value multiplied by other from left.
It follows (g.left_act(h))[x] == g[h[x]]
.
inverse
Returns the inverse element.
inv
Alias of inverse.
sign
Returns the sign of self.
conjugate(g)
Returns the conjugate by g: g * self * g.inv
.
decompose_cyclic
Returns the array of arrays of cyclic permutations. This is the inverse of ::cyclic2perm(c, n).
to_map
Returns the Map object of self.
decompose_transposition
Decompose into the array of the transpositions.