# Difference between revisions of "Crossed homomorphism"

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− | ''of a group $ G $ | + | ''of a group $ G $ into a group $ \Gamma $ with group of operators $ G $'' |

− | into a group $ \Gamma $ | ||

− | with group of operators $ G $'' | ||

A mapping $ \phi : G \rightarrow \Gamma $ | A mapping $ \phi : G \rightarrow \Gamma $ | ||

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If $ G $ | If $ G $ | ||

acts trivially on $ \Gamma $, | acts trivially on $ \Gamma $, | ||

− | then crossed homomorphisms are just ordinary homomorphisms. Crossed homomorphisms are also called $ 1 $- | + | then crossed homomorphisms are just ordinary homomorphisms. Crossed homomorphisms are also called $ 1 $-cocycles of $ G $ |

− | cocycles of $ G $ | + | with values in $ \Gamma $ (see [[Non-Abelian cohomology|Non-Abelian cohomology]]). Every element $ \gamma \in \Gamma $ |

− | with values in $ \Gamma $( | + | defines a crossed homomorphism $ \phi ( a) = \gamma ^ {-1} ( a \gamma ) $ ($ a \in G $), |

− | see [[Non-Abelian cohomology|Non-Abelian cohomology]]). Every element $ \gamma \in \Gamma $ | ||

− | defines a crossed homomorphism $ \phi ( a) = \gamma ^ {-} | ||

− | $ a \in G $), | ||

called a principal crossed homomorphism, or cocycle cohomologous to $ e $. | called a principal crossed homomorphism, or cocycle cohomologous to $ e $. | ||

A mapping $ \phi : G \rightarrow \Gamma $ | A mapping $ \phi : G \rightarrow \Gamma $ | ||

is a crossed homomorphism if and only if the mapping $ \rho $ | is a crossed homomorphism if and only if the mapping $ \rho $ | ||

of $ G $ | of $ G $ | ||

− | into the holomorph of $ \Gamma $( | + | into the holomorph of $ \Gamma $ (cf. [[Holomorph of a group|Holomorph of a group]]) given by $ \rho ( a) = ( \phi ( a) , \sigma ( a) ) $, |

− | 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 $ | where $ \sigma : G \rightarrow \mathop{\rm Aut} \Gamma $ | ||

is the homomorphism defining the $ G $ | is the homomorphism defining the $ G $ | ||

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of $ G $ | of $ G $ | ||

by affine transformations of $ V $. | by affine transformations of $ V $. | ||

− | The set $ \phi ^ {-} | + | The set $ \phi ^ {-1} ( e) \subset G $ |

is called the kernel of the crossed homomorphism $ \phi $; | is called the kernel of the crossed homomorphism $ \phi $; | ||

it is always a subgroup of $ G $. | it is always a subgroup of $ G $. |

## Latest revision as of 08:28, 18 February 2022

*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 $.

#### Comments

#### 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=46557