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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271401.png" /> into a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271402.png" /> with group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271403.png" />''
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271404.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271405.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271406.png" /> acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271407.png" />, then crossed homomorphisms are just ordinary homomorphisms. Crossed homomorphisms are also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c0271409.png" />-cocycles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714010.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714011.png" /> (see [[Non-Abelian cohomology|Non-Abelian cohomology]]). Every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714012.png" /> defines a crossed homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714014.png" />), called a principal crossed homomorphism, or cocycle cohomologous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714015.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714016.png" /> is a crossed homomorphism if and only if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714018.png" /> into the holomorph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714019.png" /> (cf. [[Holomorph of a group|Holomorph of a group]]) given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714021.png" /> is the homomorphism defining the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714022.png" /> action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714023.png" />, is a homomorphism. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714024.png" /> is a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714025.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714026.png" />, then any crossed homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714027.png" /> defines a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714029.png" /> by affine transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714030.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714031.png" /> is called the kernel of the crossed homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714032.png" />; it is always a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027140/c02714033.png" />.
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''of a group  $  G $ into a group  $  \Gamma $ with group of operators  $  G $''
  
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A mapping  $  \phi :  G \rightarrow \Gamma $
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satisfying the condition  $  \phi ( a b ) = \phi ( a) ( a \phi ( b) ) $.
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If  $  G $
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acts trivially on  $  \Gamma $,
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then crossed homomorphisms are just ordinary homomorphisms. Crossed homomorphisms are also called  $  1 $-cocycles of  $  G $
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with values in  $  \Gamma $ (see [[Non-Abelian cohomology|Non-Abelian cohomology]]). Every element  $  \gamma \in \Gamma $
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defines a crossed homomorphism  $  \phi ( a) = \gamma  ^ {-1} ( a \gamma ) $ ($  a \in G $),
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called a principal crossed homomorphism, or cocycle cohomologous to  $  e $.
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A mapping  $  \phi :  G \rightarrow \Gamma $
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is a crossed homomorphism if and only if the mapping  $  \rho $
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of  $  G $
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into the holomorph of  $  \Gamma $ (cf. [[Holomorph of a group|Holomorph of a group]]) given by  $  \rho ( a) = ( \phi ( a) , \sigma ( a) ) $,
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where  $  \sigma :  G \rightarrow  \mathop{\rm Aut}  \Gamma $
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is the homomorphism defining the  $  G $
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action on  $  \Gamma $,
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is a homomorphism. For example, if  $  \sigma $
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is a linear representation of  $  G $
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in a vector space  $  V $,
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then any crossed homomorphism  $  \phi :  G \rightarrow V $
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defines a representation  $  \rho $
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of  $  G $
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by affine transformations of  $  V $.
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The set  $  \phi  ^ {-1} ( e) \subset  G $
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is called the kernel of the crossed homomorphism  $  \phi $;
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it is always a subgroup of  $  G $.
  
 
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====References====
 
====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>
 
<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>

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=18894
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article