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Difference between revisions of "Crossed modules"

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A concept arising from the concept of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271501.png" />-module (see [[Module|Module]]). A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271502.png" /> (not necessarily Abelian) with a group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271503.png" /> and homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271504.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271505.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271506.png" />,
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A concept arising from the concept of a $G$-module (see [[Module|Module]]). A group $M$ (not necessarily Abelian) with a group of operators $G$ and homomorphism $f : M \rightarrow G$ such that for any $g \in G$ and all $x,y \in M$,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271507.png" /></td> </tr></table>
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$$
 
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f(gx) = g f(x) g^{-1}, \ \ \  f(x)y = x y x^{-1} \ .
is called a crossed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c0271509.png" />-module. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c02715010.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c02715011.png" />-module (i.e. its underlying group is Abelian) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c02715012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027150/c02715013.png" />).
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is called a crossed $(G,f)$-module. $M$ is a $G$-module (i.e. its underlying group is Abelian) if and only if $f$ is constant, equal to $e \in G$.

Latest revision as of 13:46, 12 December 2013

A concept arising from the concept of a $G$-module (see Module). A group $M$ (not necessarily Abelian) with a group of operators $G$ and homomorphism $f : M \rightarrow G$ such that for any $g \in G$ and all $x,y \in M$, $$ f(gx) = g f(x) g^{-1}, \ \ \ f(x)y = x y x^{-1} \ . $$ is called a crossed $(G,f)$-module. $M$ is a $G$-module (i.e. its underlying group is Abelian) if and only if $f$ is constant, equal to $e \in G$.

How to Cite This Entry:
Crossed modules. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Crossed_modules&oldid=11880
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article