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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  ^ {-} 1 ( 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  ^ {-} 1 ( e) \subset  G $
+
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=52060
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article