construct a permutation class - Maple Help

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Perm - construct a permutation class

Calling Sequence

Perm( L )

Perm( LL )

Parameters

L

-

list(posint) : a list of positive integers that forms a permutation of 1..n, for some n

LL

-

 list(list(posint)) : a list of lists of positive integers representing disjoint cycles

Description

• 

A permutation is a bijective mapping from the set1,2,,n to itself, for some positive integer n.

• 

The set of all such permutations forms the symmetric group of degree n, and subgroups of symmetric groups are permutation groups.

• 

Permutations are typically represented as products of disjoint cycles, each of which is an orbit of the permutation.  This is a list of the formc1,c2,,ck in which each ci is itself a listi1,i2,,im representing a cycle of the formi1i2imi1 .

• 

The Perm constructor creates a permutation, given a specification of its disjoint cycle structure in the form of a list of lists.  You can also use a permutation list, which is just the representation of the permutation as a list L of points in which L[ i ] specifies the image of i under the permutation.

• 

The Permutation Operations in GroupTheory page lists commands that operate on permutation objects and are part of the GroupTheory package.

• 

Note that the non-commutative multiplication operator . can be used to multiply permutations.

Examples

a := Perm( [ [ 1, 2 ], [ 3, 4, 5 ] ] );

a:=1,23,4,5

(1)

a[ 1 ]; # the image of 1 under a

2

(2)

a[ 2 ]; # the image of 2 under a

1

(3)

a[ 3 ]; # the image of 3 under a

4

(4)

a[ 4 ]; # the image of 4 under a

5

(5)

a[ 5 ]; # the image of 5 under a

3

(6)

b := Perm( [ [ 1, 3 ], [ 2, 6 ] ] );

b:=1,32,6

(7)

In the following examples, the PermDegree() and PermProduct() commands are part of the GroupTheory package. They operate on permutation objects constructed by Perm.

with( GroupTheory ):

PermDegree( a );

5

(8)

PermDegree( b );

6

(9)

PermProduct( a, b );

1,6,2,3,4,5

(10)

See Also

GroupTheory, Permutation Operations in GroupTheory


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