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42 Permutations

GAP offers a data type permutation to describe the elements of permutation groups.

The points on which permutations in GAP act are the positive integers up to a certain architecture dependent limit, and the image of a point \(i\) under a permutation \(p\) is written \(i^p\), which is expressed as \(i\)^\(p\) in GAP. (This action is also implemented by the function OnPoints (41.2-1).) If \(i\)^\(p\) is different from \(i\), we say that \(i\) is moved by \(p\), otherwise it is fixed. Permutations in GAP are entered and displayed in cycle notation, such as (1,2,3)(4,5).

The preimage of the point \(i\) under the permutation \(p\) can be computed as \(i\)/\(p\), without constructing the inverse of \(p\).

For arithmetic operations for permutations and their precedence, see 31.12.

In the names of the GAP functions that deal with permutations, the word "Permutation" is usually abbreviated to "Perm", to save typing. For example, the category test function for permutations is IsPerm (42.1-1).

42.1 IsPerm (Filter)

Internally, GAP stores a permutation as a list of the \(d\) images of the integers \(1, \ldots, d\), where the "internal degree" \(d\) is the largest integer moved by the permutation or bigger. When a permutation is read in in cycle notation, \(d\) is always set to the largest moved integer, but a bigger \(d\) can result from a multiplication of two permutations, because the product is not shortened if it fixes \(d\). The images are stored all as \(16\)-bit integers or all as \(32\)-bit integers, depending on whether \(d \leq 65536\) or not. For example, if \(m\geq 65536\), the permutation \((1, 2, \ldots, m)\) has internal degree \(d=m\) and takes \(4m\) bytes of memory for storage. But --- since the internal degree is not reduced --- this means that the identity permutation () calculated as \((1, 2, \ldots, m) * (1, 2, \ldots, m)^{{-1}}\) also takes \(4m\) bytes of storage. It can take even more because the internal list has sometimes room for more than \(d\) images.

The operation RestrictedPerm (42.5-4) reduces the storage degree of its result and therefore can be used to save memory if intermediate calculations in large degree result in a small degree result.

Permutations do not belong to a specific group. That means that one can work with permutations without defining a permutation group that contains them.

gap> (1,2,3);
(1,2,3)
gap> (1,2,3) * (2,3,4);
(1,3)(2,4)
gap> 17^(2,5,17,9,8);
9
gap> OnPoints(17,(2,5,17,9,8));
9

The operation Permuted (21.20-18) can be used to permute the entries of a list according to a permutation.

42.1-1 IsPerm

Each permutation in GAP lies in the category IsPerm. Basic operations for permutations are LargestMovedPoint (42.3-2), multiplication of two permutations via *, and exponentiation ^ with first argument a positive integer \(i\) and second argument a permutation \(\pi\), the result being the image \(i^{\pi}\) of the point \(i\) under \(\pi\).

42.1-2 IsPermCollection

are the categories for collections of permutations and collections of collections of permutations, respectively.

42.1-3 PermutationsFamily

is the family of all permutations.

42.2 Comparison of Permutations

42.2-1 \=

Two permutations are equal if they move the same points and all these points have the same images under both permutations.

The permutation p1 is smaller than p2 if p1 \(\neq\) p2 and \(i^{{\textit{p1}}} < i^{{\textit{p2}}}\), where \(i\) is the smallest point with \(i^{{\textit{p1}}} \neq i^{{\textit{p2}}}\). Therefore the identity permutation is the smallest permutation, see also Section 31.11.

Permutations can be compared with certain other GAP objects, see 4.12 for the details.

gap> (1,2,3) = (2,3,1);
true
gap> (1,2,3) * (2,3,4) = (1,3)(2,4);
true
gap> (1,2,3) < (1,3,2);      # 1^(1,2,3) = 2 < 3 = 1^(1,3,2)
true
gap> (1,3,2,4) < (1,3,4,2);  # 2^(1,3,2,4) = 4 > 1 = 2^(1,3,4,2)
false

42.2-2 DistancePerms

returns the number of points for which perm1 and perm2 have different images. This should always produce the same result as NrMovePoints(perm1/perm2) but some methods may be much faster than this form, since no new permutation needs to be created.

42.2-3 SmallestGeneratorPerm

is the smallest permutation that generates the same cyclic group as the permutation perm. This is very efficient, even when perm has large order.

gap> SmallestGeneratorPerm( (1,4,3,2) );
(1,2,3,4)

42.3 Moved Points of Permutations

42.3-1 SmallestMovedPoint

is the smallest positive integer that is moved by perm if such an integer exists, and infinity (18.2-1) if perm is the identity. For C a collection or list of permutations, the smallest value of SmallestMovedPoint for the elements of C is returned (and infinity (18.2-1) if C is empty).

42.3-2 LargestMovedPoint

For a permutation perm, this attribute contains the largest positive integer which is moved by perm if such an integer exists, and 0 if perm is the identity. For C a collection or list of permutations, the largest value of LargestMovedPoint for the elements of C is returned (and 0 if C is empty).

42.3-3 MovedPoints

is the proper set of the positive integers moved by at least one permutation in the collection C, respectively by the permutation perm.

42.3-4 NrMovedPoints

is the number of positive integers that are moved by perm, respectively by at least one element in the collection C. (The actual moved points are returned by MovedPoints (42.3-3).)

gap> SmallestMovedPointPerm((4,5,6)(7,2,8));
2
gap> LargestMovedPointPerm((4,5,6)(7,2,8));
8
gap> NrMovedPointsPerm((4,5,6)(7,2,8));
6
gap> MovedPoints([(2,3,4),(7,6,3),(5,47)]);
[ 2, 3, 4, 5, 6, 7, 47 ]
gap> NrMovedPoints([(2,3,4),(7,6,3),(5,47)]);
7
gap> SmallestMovedPoint([(2,3,4),(7,6,3),(5,47)]);
2
gap> LargestMovedPoint([(2,3,4),(7,6,3),(5,47)]);
47
gap> LargestMovedPoint([()]);
0

42.4 Sign and Cycle Structure

42.4-1 SignPerm

The sign of a permutation perm is defined as \((-1)^k\) where \(k\) is the number of cycles of perm of even length.

The sign is a homomorphism from the symmetric group onto the multiplicative group \(\{ +1, -1 \}\), the kernel of which is the alternating group.

42.4-2 CycleStructurePerm

is the cycle structure (i.e. the numbers of cycles of different lengths) of the permutation perm. This is encoded in a list \(l\) in the following form: The \(i\)-th entry of \(l\) contains the number of cycles of perm of length \(i+1\). If perm contains no cycles of length \(i+1\) it is not bound. Cycles of length 1 are ignored.

gap> SignPerm((1,2,3)(4,5));
-1
gap> CycleStructurePerm((1,2,3)(4,5,9,7,8));
[ , 1,, 1 ]

42.5 Creating Permutations

42.5-1 ListPerm

is a list \(l\) that contains the images of the positive integers under the permutation perm. That means that \(l\)[\(i\)] \(= i\)^perm, where \(i\) lies between 1 and the largest point moved by perm (see LargestMovedPoint (42.3-2)).

An optional second argument specifies the length of the desired list.

42.5-2 PermList

is the permutation \(\pi\) that moves points as described by the list list. That means that \(i^{\pi} =\) list[\(i\)] if \(i\) lies between \(1\) and the length of list, and \(i^{\pi} = i\) if \(i\) is larger than the length of the list list. PermList will return fail if list does not define a permutation, i.e., if list is not dense, or if list contains a positive integer twice, or if list contains an integer not in the range [ 1 .. Length( list ) ]. If list contains non-integer entries an error is raised.

42.5-3 MappingPermListList

Let src and dst be lists of positive integers of the same length, such that neither may contain an element twice. MappingPermListList returns a permutation \(\pi\) such that src[\(i\)]^\(\pi =\) dst[\(i\)]. The permutation \(\pi\) fixes all points larger than the maximum of the entries in src and dst. If there are several such permutations, it is not specified which of them MappingPermListList returns.

42.5-4 RestrictedPerm

RestrictedPerm returns the new permutation that acts on the points in the list list in the same way as the permutation perm, and that fixes those points that are not in list. The resulting permutation is stored internally of degree given by the maximal entry of list. list must be a list of positive integers such that for each \(i\) in list the image \(i\)^perm is also in list, i.e., list must be the union of cycles of perm.

RestrictedPermNC does not check whether list is a union of cycles.

gap> ListPerm((3,4,5));
[ 1, 2, 4, 5, 3 ]
gap> PermList([1,2,4,5,3]);
(3,4,5)
gap> MappingPermListList([2,5,1,6],[7,12,8,2]);
(1,8,5,12,11,10,9,6,2,7,4,3)
gap> RestrictedPerm((1,2)(3,4),[3..5]);
(3,4)

42.5-5 AsPermutation

Returns: A permutation or fail.

Partial permutations and transformations which define permutations (mathematically) can be converted into GAP permutations using AsPermutation; see Chapters 53 and 54 for more details about transformations and partial permutations.

for partial permutations

If the partial permutation f is a permutation of its image, then AsPermutation returns this permutation. If f does not permute its image, then fail is returned.

for transformations

A transformation is a permutation if and only if its rank equals its degree. If a transformation in GAP is a permutation, then AsPermutation returns this permutation. If f is not a permutation, then fail is returned.

The function Permutation (41.9-1) can also be used to convert partial permutations and transformations into permutations where appropriate.

gap> f:=PartialPerm( [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ],
> [ 2, 7, 9, 4, 1, 10, 5, 6, 3, 8 ] );
(1,2,7,5)(3,9)(4)(6,10,8)
gap> AsPermutation(f);
(1,2,7,5)(3,9)(6,10,8)
gap> f:= PartialPerm( [ 1, 2, 3, 4, 5, 7, 8 ], [ 5, 3, 8, 1, 9, 4, 10 ] );
[2,3,8,10][7,4,1,5,9]
gap> AsPermutation(f);
fail
gap> f:=Transformation( [ 5, 8, 3, 5, 8, 6, 2, 2, 7, 8 ] );;
gap> AsPermutation(f);
fail  
gap> f:=Transformation( [ 1, 3, 6, 6, 2, 10, 2, 3, 10, 5 ] );;
gap> AsPermutation(f);
fail
gap> f:=Transformation( [ 2, 7, 9, 4, 1, 10, 5, 6, 3, 8 ] );
Transformation( [ 2, 7, 9, 4, 1, 10, 5, 6, 3, 8 ] )
gap> AsPermutation(f);
(1,2,7,5)(3,9)(6,10,8)

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