vanishing 2-cohomology for $G$-module $mathbbC^G$?

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Let $G$ be a countably infinite group.



Is it true that $H^2(G, mathbbC^G)=0$ ?



Here, $mathbbC^G$ denotes all functions from $G$ to $mathbbC$ and is treated as a $G$-module, i.e. $(gf)(g'):=f(g^-1g')$ for all $g, g'in G$ and $fin mathbbC^G$.



The second cohomology group is defined as the quotient of the set of all 2-cocycles by the subset of all 2-coboundaries.



More precisely, I am asking the following.



If $c: Gtimes Gto mathbbC^G$ is any map satisfying $c(g_1, g_2)+c(g_1g_2, g_3)=g_1c(g_2, g_3)+c(g_1, g_2g_3)$ for all $g_iin G$, can be write $c$ as $c(g_1, g_2)=b(g_1)-b(g_1g_2)+g_1b(g_2)$ for some function $b: Gto mathbbC^G$ and all $gin G$?




reference-request group-cohomology




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    up vote
    1
    down vote

    favorite
    1












    Let $G$ be a countably infinite group.



    Is it true that $H^2(G, mathbbC^G)=0$ ?



    Here, $mathbbC^G$ denotes all functions from $G$ to $mathbbC$ and is treated as a $G$-module, i.e. $(gf)(g'):=f(g^-1g')$ for all $g, g'in G$ and $fin mathbbC^G$.



    The second cohomology group is defined as the quotient of the set of all 2-cocycles by the subset of all 2-coboundaries.



    More precisely, I am asking the following.



    If $c: Gtimes Gto mathbbC^G$ is any map satisfying $c(g_1, g_2)+c(g_1g_2, g_3)=g_1c(g_2, g_3)+c(g_1, g_2g_3)$ for all $g_iin G$, can be write $c$ as $c(g_1, g_2)=b(g_1)-b(g_1g_2)+g_1b(g_2)$ for some function $b: Gto mathbbC^G$ and all $gin G$?




    reference-request group-cohomology




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      Let $G$ be a countably infinite group.



      Is it true that $H^2(G, mathbbC^G)=0$ ?



      Here, $mathbbC^G$ denotes all functions from $G$ to $mathbbC$ and is treated as a $G$-module, i.e. $(gf)(g'):=f(g^-1g')$ for all $g, g'in G$ and $fin mathbbC^G$.



      The second cohomology group is defined as the quotient of the set of all 2-cocycles by the subset of all 2-coboundaries.



      More precisely, I am asking the following.



      If $c: Gtimes Gto mathbbC^G$ is any map satisfying $c(g_1, g_2)+c(g_1g_2, g_3)=g_1c(g_2, g_3)+c(g_1, g_2g_3)$ for all $g_iin G$, can be write $c$ as $c(g_1, g_2)=b(g_1)-b(g_1g_2)+g_1b(g_2)$ for some function $b: Gto mathbbC^G$ and all $gin G$?




      reference-request group-cohomology




      share|cite|improve this question












      Let $G$ be a countably infinite group.



      Is it true that $H^2(G, mathbbC^G)=0$ ?



      Here, $mathbbC^G$ denotes all functions from $G$ to $mathbbC$ and is treated as a $G$-module, i.e. $(gf)(g'):=f(g^-1g')$ for all $g, g'in G$ and $fin mathbbC^G$.



      The second cohomology group is defined as the quotient of the set of all 2-cocycles by the subset of all 2-coboundaries.



      More precisely, I am asking the following.



      If $c: Gtimes Gto mathbbC^G$ is any map satisfying $c(g_1, g_2)+c(g_1g_2, g_3)=g_1c(g_2, g_3)+c(g_1, g_2g_3)$ for all $g_iin G$, can be write $c$ as $c(g_1, g_2)=b(g_1)-b(g_1g_2)+g_1b(g_2)$ for some function $b: Gto mathbbC^G$ and all $gin G$?





      reference-request group-cohomology





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      asked Aug 13 at 9:54









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