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? $\endgroup$– Fernando MuroCommented Sep 17, 2014 at 20:50
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1$\begingroup$ It should say an inequality involving the dimensions of those groups, regarded as $\mathbb{F}_p$-vector spaces. $\endgroup$– Cam McLemanCommented Sep 17, 2014 at 21:10
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