Dom::PermutationGroup
-- permutation groupsDom::PermutationGroup
(n)
creates the
domain of all permutations of n elements.
Dom::PermutationGroup(n)
n |
- | positive integer |
The domain element
Dom::PermutationGroup
(n)(l)
represents the
bijective mapping of the first n positive integers that maps
the integer i to l[i]
, for 1 ≤i
≤n.
Dom::PermutationGroup
(n)(l)
l |
- | list or array consisting of the first n integers in some order. |
Cat::Group
the identical mapping of the set {1, ..., n } to itself.
_mult(dom a1...)
a1*a2*...*ak
of permutations is defined to
be the mapping that assigns, to every integer i
between
1 and n, the integer a1(a2(...ak(i)
...))
._mult
._invert(dom a)
b
such that
a*b
is the identity mapping._invert
.func_call(dom a, integer i)
(...)
, i.e.
it may be called in the form a(i)
.a
at i
,
i.e., the integer that i
is mapped to by the permutation
a
; i
must be an integer between 1
and n.cycles(dom a)
a
. A
cycle representation is a list [orbit1, ..., orbitk]
; each
of the orbits is a list of integers of the form [i, a(i),
a(a(i)), ...]
with just as many elements such that
i
does not occur in it for a second time; and each integer
between 1 and n appears in exactly one of the
orbits.order(dom a)
a
is defined to be the least positive
integer k
for which ak is the
identity.inversions(dom a)
random()
convert(any x)
x
into a permutation.
This is only possible if x
is a list or an array in which
each of the integers 1 through n occurs exactly
once.convert_to(dom a, any
T)
a
into type T
.
Currently, only a conversion into a list of type DOM_LIST
is possible.expr(dom a)
a
.Consider the group of permutations of the first seven positive integers:
>> G:=Dom::PermutationGroup(7)
Dom::PermutationGroup(7)
We enter an element by providing the image of 1, 2, etc. under the permutation.
>> a:=G([2,4,6,1,3,5,7])
[2, 4, 6, 1, 3, 5, 7]
>> a(3)
6
Ax::canonicalRep