MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

The restriction $res^G_H:H^i(G)\rightarrow H^i(H)$ is nonzero for an infinite number of values of $i>0$.
As a corollary, for any prime $p$ dividing $|G|$, the $p$-primary component $H^i(G)_{(p)}$ is nonzero for an infinite number of values of $i>0$.
He actually proves this in a more general case, where $G$ is a compact Lie group and $H$ is a closed subgroup (defining group cohomology with the classifying spaces $BG$ and $BH$. BH$). The proof uses basic cohomological and Lie group principles. 1 Yes, Richard Swan is the first to prove it, The Nontriviality of the Restriction Map in the Cohomology of Groups (1959). He shows that the restriction$res^G_H:H^i(G)\rightarrow H^i(H)$is nonzero for an infinite number of values of$i>0$. As a corollary, for any prime$p$dividing$|G|$,$H^i(G)_{(p)}$is nonzero for an infinite number of values of$i>0$. He actually proves this in a more general case, where$G$is a compact Lie group and$H$is a closed subgroup (defining group cohomology with the classifying spaces$BG$and$BH\$. The proof uses basic cohomological and Lie group principles.