View source for Crossed homomorphism

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''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|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|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 $.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,   "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lang,   "Rapport sur la cohomologie des groupes" , Benjamin  (1966)</TD></TR></table>