# 7.108 Mapping Functions for Group Homomorphisms

This section describes how the mapping functions defined in chapter Mappings are implemented for group homomorphisms. Those functions not mentioned here are implemented by the default functions described in the respective sections.

IsInjective( hom )

The group homomorphism hom is injective if the kernel of hom KernelGroupHomomorphism( hom ) (see KernelGroupHomomorphism) is trivial.

IsSurjective( hom )

The group homomorphism hom is surjective if the size of the image Size( Image( hom ) ) (see Image and below) is equal to the size of the range Size( hom.range ).

hom1 = hom2

The two group homomorphisms hom1 and hom2 are equal if the have the same source and range and if the images of the generators of the source under hom1 and hom2 are equal.

hom1 < hom2

By definition hom1 is smaller than hom2 if either the source of hom1 is smaller than the source of hom2, or, if the sources are equal, if the range of hom1 is smaller than the range of hom2, or, if sources and ranges are equal, the image of the smallest element x of the source for that the images are not equal under hom1 is smaller than the image under hom2. Therefor GroupHomomorphismOps.< first compares the sources and the ranges. For group homomorphisms with equal sources and ranges only the images of the smallest irredundant generating system are compared. A generating system g_1, g_2, ..., g_n is called irredundant if no g_i lies in the subgroup generated by g_1, ..., g_{i-1}. The smallest irredundant generating system is simply the smallest such generating system with respect to the lexicographical ordering.

Image( hom ) Image( hom, H )
Images( hom, H )

The image of a subgroup under a group homomorphism is computed by computing the images of a set of generators of the subgroup, and the result is the subgroup generated by those images.

PreImages( hom, elm )

The preimages of an element under a group homomorphism are computed by computing a representative with PreImagesRepresentative( hom, elm ) and the result is the coset of Kernel( hom ) containing this representative.

PreImage( hom ) PreImage( hom, H )
PreImages( hom, H )

The preimages of a subgroup under a group homomorphism are computed by computing representatives of the preimages of all the generators of the subgroup, adding the generators of the kernel of hom, and the result is the subgroup generated by those elements.

Look under IsInjective, IsSurjective, equality, ordering, Image, Images, PreImage, and PreImages in the index to see for which group homomorphisms these functions are overlaid.

GAP 3.4.4
April 1997