# Crossed homomorphism

*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 .

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#### References

[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: http://encyclopediaofmath.org/index.php?title=Crossed_homomorphism&oldid=18894