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Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed homomorphism $f:\,\pi\rightarrow D$ (i.e., a function such that $f\left(\sigma\tau\right)=f\left(\sigma\right)^{\tau}+f\left(\tau\right)$ for every $\sigma,\tau\in\pi$ , where $\tau$ acts on $f\left(\sigma\right)$ ). Is it true that $nf$ is a principal homomorphism? Thanks in advance for any help or suggestion.

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  • $\begingroup$ What do you conjecture? Is it? Did you try out any examples? Did you try proving it? Please share your thoughts with us first. $\endgroup$ – M. Vinay Jun 3 '16 at 14:41
  • $\begingroup$ I want to prove that $nf$ is a principal homomorphism. It seems to me that you cannot prove it directly, but using in some way the correspondence between conjugacy classes of complements of $D$ in the semidirect product $\pi\times_{\varphi}D$ ($\varphi$ is the action of $\pi$ on $D$ ) and $H^{1}\left(\pi,D\right)$ . I suppose that the fundamental hypothesis is the finiteness of the group $\pi$ , but I can be wrong. $\endgroup$ – Bimbumbam Jun 3 '16 at 14:57
  • $\begingroup$ It is known that for any finite group G and any G-module M, the cohomology groups H^i (G, M) , for i > 0, are killed by the order of G. This is a consequence of the properties of the maps Res and Cores. See e.g. Serre's "Local Fields", chapter VIII, coroll. 1 of propos. 4. $\endgroup$ – nguyen quang do Jun 4 '16 at 6:47
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Actually this question was already asked in mathstack exchange. The answer is here first homology group with coefficients in divisible group

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  • $\begingroup$ Okay. You could upvote that answer (and question too, maybe) if it was helpful to you. $\endgroup$ – M. Vinay Jun 3 '16 at 16:03

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