# 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 $G$.