 # 3 Transformations

A transformation of degree n is a map from the set {1, ... , n} into itself. Thus a transformation alpha of degree n associates a positive integer i^alpha less than or equal to n to each number i between 1 and n.

The degree of a transformation may not be larger than 2^{28}-1 which is (currently) the highest index that can be accessed in a list.

Special cases of transformations are permutations (see chapter "Permutations"). However, a permutation must be converted to a transformation before most of the functions in this chapter are applicable.

The product of transformations is defined via composition of maps. Here transformations are multiplied in such a way that they act from the right on the set {1, ... , n}. That is, the product of the transformations alpha and beta of degree n is defined by [ i^(alphabeta) = (i^alpha)^betaquadmboxfor all i = 1, ... ,n. ] With respect to this multiplication the set of all transformations of degree n forms a monoid: the full transformation monoid of degree n (see chapter Transformation Monoids).

Each transformation of degree n is considered an element of the full transformation monoid of degree n although it is not necessary to construct a full transformation monoid before working with transformations. But you can only multiply two transformations if they have the same degree. You can, however, multiply a transformation of degree n by a permutation of degree n.

Transformations are entered and displayed by giving their lists of images as an argument to the function `Transformation`.

```    gap> Transformation( [ 3, 3, 4, 2, 5 ] );
Transformation( [ 3, 3, 4, 2, 5 ] )
gap> Transformation( [ 3, 3, 2 ] ) * Transformation( [ 1, 2, 1 ] );
Transformation( [ 1, 1, 2 ] )```

This chapter describes functions that deal with transformations. The first sections describe the representation of a transformation in GAP (see More about Transformations) and how a transformation is constructed as a GAP object (see Transformation). The next sections describe the comparisons and the operations which are available for Operations for Transformations). There are a function to test whether an arbitrary object is a transformation (see IsTransformation) and a function to construct the identity transformation of a given degree (see IdentityTransformation). Then there are functions that compute Kernel of a Transformation). Finally, there are a function that converts a permutation to a transformation (see TransPerm) and a function that, if possible converts a transformation to a permutation (see PermTrans).

The functions described here are in the file `"transfor.g"`.

### Subsections   [Index]

Version 2.4 (May 1998) 