# 3.4 Comparisons of Transformations

`tr1 = tr2`
`tr1 < tr2`

The equality operator `=` applied to two transformations tr1 and tr2 evaluates to `true` if the two transformations are equal and to `false` otherwise. The inequality operator `<` applied to two transformations tr1 and tr2 evaluates to `true` if the two transformations are not equal and to `false` otherwise. A transformation can also be compared to any other object that is not a transformation, of course they are never equal. Two transformations are considered equal if and only if their image lists are equal as lists. In particular, equal transformations must have the same degree.

```    gap> Transformation( [ 1, 2, 3, 4 ] ) = IdentityTransformation( 4 );
true
gap> Transformation( [ 1, 4, 4, 2 ] ) =
> Transformation( [ 1, 4, 4, 2, 5 ] );
false```

`tr1 < tr2`
`tr1 <= tr2`
`tr1 tr2`
`tr1 = tr2`

The operators `<`, `<=`, , and `=` evaluate to `true` if the transformation tr1 is less than, less than or equal to, greater than, or greater than or equal to the transformation tr2, and to `false` otherwise.

Let tr1 and tr2 be two transformations that are not equal. Then tr1 is considered smaller than tr2 if and only if the list of images of tr1 is (lexicographically) smaller than the list of images of tr2. Note that this way the smallest transformation of degree n is the transformation that maps every point to 1.

You can also compare transformations with objects of other types. Here any object that is not a transformation will be considered smaller than any transformation.

Version 2.4 (May 1998)