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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 1959):

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.
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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.