As the title indicates, my question has to do with something rather simple. So, in Kenneth's Brown book "Cohomology of finite groups" at pg.84-85 and in particular Theorem 10.3 and Proposition 10.4, there is one thing that I don' t get. Whilst, at the former for $H$ a normal $p$-Sylow subgroup of $G$ we found that the $p$-component of a $G$-module $M$ is $H^{n}(G ; M)_{(p)} \simeq H^{n}(H ; M)^{G/H}$, $\forall n > 0$, the next proposition says that if $H$ is a normal subgroup of finite index in $G$ then $H^{*}(G ; M) \simeq H^{*}(H ; M)^{G/H}$. If we assume that at the latter the group $H$ is $p$-Sylow then what's the difference between those things? Why does he change the notation? I can't see the why.
Thank you!
In Proposition 10.4 there's an extra hypothesis that $[G:H]$ is invertible in $M$, which implies that $H^n(G,M)_{(p)}=H^n(G,M)$ for $n>0$ if $H$ is a Sylow $p$-subgroup.