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As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients in $F_p$. My question: Do we know any inequalities between $\operatorname{dim}H^{i+1}(G, F_p)$ and $\operatorname{dim}H^{i}(G, F_p)$ for a $p$-group $G$ for all $i$ or say at least for $i=2$?

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  • $\begingroup$ What does inequality between abelian groups mean? $\endgroup$ Sep 17, 2014 at 20:50
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    $\begingroup$ It should say an inequality involving the dimensions of those groups, regarded as $\mathbb{F}_p$-vector spaces. $\endgroup$ Sep 17, 2014 at 21:10

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