Dom::SymmetricGroup --
symmetric groups
IntroductionDom::SymmetricGroup(n) creates the symmetric group of order n, that
is, the domain of all the permutations of {1,...,n}
elements.
Domain
Dom::SymmetricGroup(n)
Parametersn | - | positive integer |
DetailsDom::SymmetricGroup inherits from all the methods and entries defined in
its super domain Dom::PermutationGroup.
IntroductionThe domain element Dom::SymmetricGroup(n)(l) represents the bijective
mapping of the first n positive integers that maps the integer
i to l[i], for 1 ≤i ≤n.
Creating Elements
Dom::SymmetricGroup(n)(l)
Parametersl | - | list or array consisting of the first n integers in some order. |
Cat::Group
Related Domainsthe 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; 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)
sign(dom a)a. The sign
of a permutation is defined to be 1 if its number of inversions is
even, and -1 otherwise.
random()
allElements()
size()
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.
Example 1Consider the group of permutations of the first seven positive integers:
>> G := Dom::SymmetricGroup(7)
Dom::SymmetricGroup(7)
We create an element of G by providing the image of 1,
2, etc.:
>> a:=G([2,4,6,1,3,5,7])
[2, 4, 6, 1, 3, 5, 7]
>> a(3)
6
Super-DomainAx::canonicalRep
MuPAD Combinat, an open source algebraic combinatorics package