Crossed homomorphism

of a group $ G $ into a group $ \Gamma $ with group of operators $ G $

A mapping $ \phi : G \rightarrow \Gamma $ satisfying the condition $ \phi ( a b ) = \phi ( a) ( a \phi ( b) ) $. If $ G $ acts trivially on $ \Gamma $, then crossed homomorphisms are just ordinary homomorphisms. Crossed homomorphisms are also called $ 1 $-cocycles of $ G $ with values in $ \Gamma $ (see Non-Abelian cohomology). Every element $ \gamma \in \Gamma $ defines a crossed homomorphism $ \phi ( a) = \gamma ^ {-1} ( a \gamma ) $ ($ a \in G $), called a principal crossed homomorphism, or cocycle cohomologous to $ e $. A mapping $ \phi : G \rightarrow \Gamma $ is a crossed homomorphism if and only if the mapping $ \rho $ of $ G $ into the holomorph of $ \Gamma $ (cf. Holomorph of a group) given by $ \rho ( a) = ( \phi ( a) , \sigma ( a) ) $, where $ \sigma : G \rightarrow \mathop{\rm Aut} \Gamma $ is the homomorphism defining the $ G $ action on $ \Gamma $, is a homomorphism. For example, if $ \sigma $ is a linear representation of $ G $ in a vector space $ V $, then any crossed homomorphism $ \phi : G \rightarrow V $ defines a representation $ \rho $ of $ G $ by affine transformations of $ V $. The set $ \phi ^ {-1} ( e) \subset G $ is called the kernel of the crossed homomorphism $ \phi $; it is always a subgroup of $ G $.

References

[a1]	S. MacLane, "Homology" , Springer (1963)
[a2]	S. Lang, "Rapport sur la cohomologie des groupes" , Benjamin (1966)