GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/PermDegree

GroupTheory

  

PermApply

  

apply a permutation to a point

  

PermDegree

  

compute the degree of a permutation

  

PermOrder

  

compute the order of a permutation

  

PermParity

  

compute the parity of a permutation

  

PermProduct

  

compute the product of two permutations

  

PermInverse

  

compute the inverse of a permutation

  

PermPower

  

compute the power of a permutation

  

PermLeftQuotient

  

compute the left quotient of two permutations

  

PermRightQuotient

  

compute the right quotient of two permutations

  

PermConjugate

  

compute the conjugate of two permutations

  

PermCommutator

  

compute the commutator of two permutations

  

PermCycleType

  

compute the cycle type of a permutation

  

PermFixed

  

compute the set of fixed points of a permutation

  

PermSupport

  

compute the support of a permutation

  

PermCompare

  

compare two permutations

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

PermApply( n, p )

PermDegree( p )

PermOrder( p )

PermIsInvolution( p )

PermParity( p )

PermProduct( p, q )

PermInverse( p )

PermPower( p, n )

PermLeftQuotient( p, q )

PermRightQuotient( p, q )

PermConjugate( p, q )

PermCommutator( p, q )

PermCycleType( p )

PermFixed( p )

PermSupport( p )

PermRestrict( p, dom )

PermShift( p, n )

PermCompare( p, q )

PermAltSplit( p )

Parameters

p, q

-

Perm : permutations

n

-

integer : an integer

dom

-

{set,list}(posint) : set or list of positive integers

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 image of a point i under a permutation p is returned by the indexed expression p[ i ].  You can also use the procedure PermApply for this purpose. The expression PermApply( p, i ) returns the image of i under the permutation p.  The PermApply command is primarily intended for use with procedures such as map and map2.

• 

The procedures described in this help page operate on permutation objects returned by the Perm constructor, and are part of the GroupTheory package (whereas Perm itself is global).

• 

The degree of a permutation is the number of points upon which it acts. The PermDegree( p ) command returns the degree of the permutation p.

• 

The product of two permutations a and b can be computed by using the PermProduct( a, b ) command.  Note that permutation multiplication is non-commutative; that is, in general, PermProduct( a, b ) and PermProduct( b, a ) are different. You can also compute the product of the permutations a and b by using the non-commutative multiplication operator `.`, as a . b.

• 

The inverse of the permutation p is computed using the command PermInverse( p ).  It can also be computed using the syntax p^(-1).

• 

An integral power of a permutation p is computed using the command PermPower( p, n ), which computes the n-th power of p, defined in the natural way. You can also use the syntax p^n.

• 

The PermOrder( p ) command computes the order of a permutation p. That is the least positive integer k for which p^k is the identity.

• 

The identity permutation can be constructed using the expression Perm( [] ).

• 

Every permutation can be written as a product of transpositions, permutations which interchange exactly two points. Although this factorization is not unique, the number of transpositions in such a product is either always even or always odd. A permutation is said to be even (respectively, odd) according to whether is has a decomposition as a product of an even (respectively, odd) number of transpositions. An even permutation is defined to have parity equal to 1, while an odd permutation is defined to have parity equal to −1. (This defines a homomorphism from the full symmetric group onto the multiplicative group −1,1, whose kernel, consisting of the even permutations, is the alternating group.) The parity of a permutation p is returned by the PermParity( p ) command.

• 

The left quotient p-1˙q of permutations p and q is computed using the PermLeftQuotient( p, q ) command.  Similarly, the command PermRightQuotient( p, q ) computes the right quotient p˙q-1 of p and q.

• 

The conjugate pq of p by q is defined to be p-1˙q˙p and may be computed by the PermConjugate( p, q ) command. The syntax p^q is also available for this computation.

• 

The commutator of permutations p and q is defined to be the permutation p-1˙q-1˙p˙q and is computed using the command PermCommutator( p, q ).

• 

The PermCycleType( p ) command returns the cycle type of the permutation p. This is defined to be the list consisting of the lengths of the cycles of p, in sorted order.

• 

A point i is fixed by a permutation p if ip=i; its complement (in the domain of the permutation) is called the support of p. The set of fixed points of a permutation p is returned by the command PermFixed( p ), and the support of p is returned by the command PermSupport( p ).

• 

If an invariant subset dom of a permutation p exists (i.e., a union of cycles of p), then the restriction of p to dom is a permutation that can be computed by using the PermRestrict( p, dom ) command.

• 

The PermShift( p, n ) command returns a permutation conjugate to p but acting on the result of adding the integer n to each point of the support of p, provided that the result is a set of positive integers.

• 

The conjugacy class of an even permutation p in the symmetric group to which it belongs may or may not split into two conjugacy classes in the alternating group. The PermAltSplit( p ) command returns the value true if this happens, and returns the value false otherwise.

• 

The PermCompare( a, b ) command compares two permutations a and b, according to the lexicographic plus degree ordering.  The PermCompare command is intended for use as the second argument in a a call to sort on a list of permutations.

Examples

aPerm1,2,3,4,5

a1,23,4,5

(1)

a1

2

(2)

a2

1

(3)

a3

4

(4)

a4

5

(5)

a5

3

(6)

bPerm1,3,2,6

b1,32,6

(7)

withGroupTheory:

PermApplya,2

1

(8)

PermDegreea

5

(9)

PermDegreeb

6

(10)

PermProducta,b

1,6,2,3,4,5

(11)

a·b

1,6,2,3,4,5

(12)

PermProductb,a

1,4,5,3,2,6

(13)

PermLeftQuotienta,b

1,6,2,3,5,4

(14)

a1·b

1,6,2,3,5,4

(15)

PermRightQuotienta,b

1,6,2,3,4,5

(16)

a·b1

1,6,2,3,4,5

(17)

PermPowera,10

3,4,5

(18)

a10

3,4,5

(19)

PermConjugatea,b

1,4,53,6

(20)

ab

1,4,53,6

(21)

PermCommutatora,b

1,2,4,6,3

(22)

PermParitya

−1

(23)

PermParityb

1

(24)

PermOrdera

6

(25)

PermOrderb

2

(26)

evalbb·b=Perm

true

(27)

PermInversea

1,23,5,4

(28)

PermInverseb

1,32,6

(29)

PermCycleTypea

2,3

(30)

PermCycleTypeb

2,2

(31)

PermSupporta

1,2,3,4,5

(32)

PermSupportb

1,2,3,6

(33)

cPermRestrictb,2,6

c2,6

(34)

PermShiftc,1

1,5

(35)

PermShiftc,3000

3002,3006

(36)

PermShiftc,3

Error, (in Perm:-extShift) invalid shift value -3

PermFixedb

4,5

(37)

aPerm1,2,3;bPerm1,2;cPerm1,2,3,4,5

a1,2,3

b1,2

c1,2,34,5

(38)

PermComparea,b

false

(39)

PermComparea,c

true

(40)

sorta,b,c,PermCompare

1,2,1,2,3,1,2,34,5

(41)

Compatibility

• 

The GroupTheory[PermApply], GroupTheory[PermDegree], GroupTheory[PermOrder], GroupTheory[PermParity], GroupTheory[PermProduct], GroupTheory[PermInverse], GroupTheory[PermPower], GroupTheory[PermLeftQuotient], GroupTheory[PermRightQuotient], GroupTheory[PermConjugate], GroupTheory[PermCommutator], GroupTheory[PermCycleType], GroupTheory[PermFixed] and GroupTheory[PermSupport] commands were introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

• 

The GroupTheory[PermCompare] command was introduced in Maple 2017.

• 

For more information on Maple 2017 changes, see Updates in Maple 2017.

See Also

GroupTheory

Perm