[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]]
unityReturns the unity.
permReturns the array which represents self
degreeReturns the degree
sizeAlias of degree.
eachIterates for each entry.
eql?(other)Returns true if self is equal to other.
==Alias of eql?.
hashReturns the hash number.
[i]Returns the number to which i is transferrd.
callAlias 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]].
inverseReturns the inverse element.
invAlias of inverse.
signReturns the sign of self.
conjugate(g)Returns the conjugate by g: g * self * g.inv.
decompose_cyclicReturns the array of arrays of cyclic permutations. This is the inverse of ::cyclic2perm(c, n).
to_mapReturns the Map object of self.
decompose_transpositionDecompose into the array of the transpositions.