37 Group Homomorphisms

A group homomorphism is a mapping from one group to another that respects multiplication and inverses. They are implemented as a special class of mappings, so in particular all operations for mappings, such as Image, PreImage, PreImagesRepresentative, KernelOfMultiplicativeGeneralMapping, Source, Range, IsInjective and IsSurjective (see chapter Mappings, in particular section Mappings that Respect Multiplication) are applicable to them.

Homomorphisms can be used to transfer calculations into isomorphic groups in another representation, for which better algoroithms are available. Section Nice Monomorphisms explains a technique how to enforce this automatically.

Homomorphisms are also used to represent group automorphisms, and section Group Automorphisms explains explains GAP's facilities to work with automorphism groups.

The penultimate section of this chapter, Searching for Homomorphisms, explains how to make GAP to search for all homomorphisms between two groups which fulfill certain specifications.

Sections

1. Creating Group Homomorphisms
2. Operations for Group Homomorphisms
3. Efficiency of Homomorphisms
4. Nice Monomorphisms
5. Group Automorphisms
6. Groups of Automorphisms
7. Calculating with Group Automorphisms
8. Searching for Homomorphisms
9. Representations for Group Homomorphisms

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