### 54 Transformations

This chapter describes functions for transformations.

A transformation in GAP is an endomorphism of a set of integers of the form { 1, ..., n }. Transformations are taken to act on the right, which defines the composition i^(α β) = (i^α)^β for i in { 1, ..., n }.

For a transformation α on the set { 1, ..., n }, we define its degree to be n, its image list to be the list [1 α, ..., n α], its image to be the image list considered as a set, and its rank to be the size of the image. We also define the kernel of α to be the equivalence relation containing the pair (i, j) if and only if i^α = j^α.

Note that unlike permutations, we do not consider unspecified points to be fixed by a transformation. Therefore multiplication is only defined on two transformations of the same degree.

#### 54.1 Functions for Transformations

##### 54.1-1 IsTransformation
 ‣ IsTransformation( obj ) ( category )
 ‣ IsTransformationCollection( obj ) ( category )

We declare it as IsMultiplicativeElementWithOne (31.14-11) since the identity automorphism of { 1, ..., n } is a multiplicative two sided identity for any transformation on the same set.

##### 54.1-2 TransformationFamily
 ‣ TransformationFamily( n ) ( function )
 ‣ TransformationType( n ) ( function )
 ‣ TransformationData( n ) ( function )

For each n > 0 there is a single family and type of transformations on n points. To speed things up, we store these in a database of types. The three functions above a then access functions. If the nth entry isn't yet created, they trigger creation as well.

For n > 0, element n of the type database is [TransformationFamily(n), TransformationType(n)]

##### 54.1-3 Transformation
 ‣ Transformation( images ) ( function )
 ‣ TransformationNC( images ) ( function )

both return a transformation with the image list images. The first version checks that the all the elements of the given list lie within the range { 1, ..., n } where n is the length of images, but for speed purposes, a non-checking version is also supplied.

##### 54.1-4 IdentityTransformation
 ‣ IdentityTransformation( n ) ( function )

returns the identity transformation of degree n.

##### 54.1-5 RandomTransformation
 ‣ RandomTransformation( n ) ( function )

returns a random transformation of degree n.

##### 54.1-6 DegreeOfTransformation
 ‣ DegreeOfTransformation( trans ) ( attribute )

returns the degree of trans.

gap> t:= Transformation([2, 3, 4, 2, 4]);
Transformation( [ 2, 3, 4, 2, 4 ] )
gap> DegreeOfTransformation(t);
5


##### 54.1-7 ImageListOfTransformation
 ‣ ImageListOfTransformation( trans ) ( attribute )

returns the image list of trans.

gap> ImageListOfTransformation(t);
[ 2, 3, 4, 2, 4 ]


##### 54.1-8 ImageSetOfTransformation
 ‣ ImageSetOfTransformation( trans ) ( attribute )

returns the image of trans as a set.

gap> ImageSetOfTransformation(t);
[ 2, 3, 4 ]


##### 54.1-9 RankOfTransformation
 ‣ RankOfTransformation( trans ) ( attribute )

returns the rank of trans.

gap> RankOfTransformation(t);
3


##### 54.1-10 KernelOfTransformation
 ‣ KernelOfTransformation( trans ) ( attribute )

returns the kernel of trans as an equivalence relation, see 33.1).

gap> KernelOfTransformation(t);
[ [ 1, 4 ], [ 2 ], [ 3, 5 ] ]


##### 54.1-11 PreimagesOfTransformation
 ‣ PreimagesOfTransformation( trans, i ) ( operation )

returns the subset of { 1, ..., n } which maps to i under trans.

gap> PreimagesOfTransformation(t, 2);
[ 1, 4 ]


##### 54.1-12 RestrictedTransformation
 ‣ RestrictedTransformation( trans, alpha ) ( operation )

The transformation trans is restricted to only those points of alpha.

##### 54.1-13 AsTransformation
 ‣ AsTransformation( O[, n] ) ( operation )
 ‣ AsTransformationNC( O, n ) ( operation )

returns the object O as a transformation. Supported objects are permutations and binary relations on points. Called with two arguments, the operation returns a transformation of degree n, signalling an error if such a representation is not possible. AsTransformationNC does not perform this check.

gap> AsTransformation((1, 3)(2, 4));
Transformation( [ 3, 4, 1, 2 ] )
gap> AsTransformation((1, 3)(2, 4), 10);
Transformation( [ 3, 4, 1, 2, 5, 6, 7, 8, 9, 10 ] )

gap> AsTransformation((1, 3)(2, 4), 3);
Error, Permutation moves points over the degree specified called from
<function>( <arguments> ) called from read-eval-loop
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;


##### 54.1-14 PermLeftQuoTransformation
 ‣ PermLeftQuoTransformation( tr1, tr2 ) ( operation )

Given transformations tr1 and tr2 with equal kernel and image, we compute the permutation induced by (tr1)^{-1} * tr2 on the set of images of tr1. If the kernels and images are not equal, an error is signaled.

##### 54.1-15 BinaryRelationTransformation
 ‣ BinaryRelationTransformation( trans ) ( operation )

returns trans when considered as a binary relation.

##### 54.1-16 TransformationRelation
 ‣ TransformationRelation( R ) ( operation )

returns the binary relation R when considered as a transformation. Only makes sense for injective binary relations over [1..n]. An error is signalled if the relation is not over [1..n], and fail if it is not injective.

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