I have a bunch of notes made from a professor about cohomology that states that
If $S$ is a $p$-Sylow subgroup of $G$ ($\vert G \vert <\infty$), then $$H^{\ast}(G,\mathbb{F}_p)\leq H^{\ast}(S,\mathbb{F}_p)$$ It is, the cohomology ring of $G$ with coefficients in $\mathbb{F}_p$ is a subring of the cohomology ring of $S$ with coefficients in $\mathbb{F}_p$
But he wrote it as a theorem without a proof. Anyone knows a reference or a way to prove this theorem?
Let $H\subseteq G$ be a subgroup. There are two standard maps on the group cohomology associated to this: $res_H^G:H^*(G, M)\to H^*(H, M|_H)$ and $tr^G_H:H^*(H, M|_H)\to H^*(G, M)$. We have the relation $tr\circ res(x)=[G:H]x$. Now we wish to show that in the case $H=S$, a Sylow-group of $G$, we have that $res_S^G$ is an injection. Letting $x\in Ker(res_S^G)$, we have $tr\circ res(x)=1=[G:S]x$, but $([G:S], ord(x))=1$ by basic group theory, so that $x=1$. Thus we can conclude using equality $H^*(S, \mathbb{F}_p|_S)=H^*(S, \mathbb{F}_p)$.