Crossed homomorphism

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of a group into a group with group of operators

A mapping satisfying the condition . If acts trivially on , then crossed homomorphisms are just ordinary homomorphisms. Crossed homomorphisms are also called -cocycles of with values in (see Non-Abelian cohomology). Every element defines a crossed homomorphism (), called a principal crossed homomorphism, or cocycle cohomologous to . A mapping is a crossed homomorphism if and only if the mapping of into the holomorph of (cf. Holomorph of a group) given by , where is the homomorphism defining the action on , is a homomorphism. For example, if is a linear representation of in a vector space , then any crossed homomorphism defines a representation of by affine transformations of . The set is called the kernel of the crossed homomorphism ; it is always a subgroup of .



[a1] S. MacLane, "Homology" , Springer (1963)
[a2] S. Lang, "Rapport sur la cohomologie des groupes" , Benjamin (1966)
How to Cite This Entry:
Crossed homomorphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article