I am trying to understand the proof of the following proposition:
Proposition Let $G$ be a finite group and let $P$ be a normal Sylow $p$-subgroup of $G$. Then for all $n$ there is an isomorphism $$ H_n(G;\mathbb{F}_p) \cong H_n(P;\mathbb{F}_p)_Q, $$ where $G$ acts trivially on $\mathbb{F}_p$ and $Q := G/P$.
Proof: The associated Hochschild-Serre spectral sequence takes the form $$ E_{s,t}^2 \cong H_s(Q;H_t(P;\mathbb{F}_p)) \implies H_n(G;\mathbb{F}_p), $$ where $s+t=n$. The groups $H_t(P;\mathbb{F}_p)$ are $\mathbb{F}_p$-vector spaces. We know that $\gcd(|Q|,p)=1$, where $|Q|$ denotes $\text{ord}(Q)$, and hence $|Q|$ is invertible in $H_t(P;\mathbb{F}_p)$. Therefore, the spectral sequence is concentrated on the line $s=0$ and we get \begin{align*} H_s(Q;H_t(P;\mathbb{F}_p)) \cong \begin{cases} 0 & s > 0, \newline H_t(P;\mathbb{F}_p)_Q & s=0. \end{cases} \end{align*} All extension problems are trivial and the result follows.
Questions
I'm not sure I understand how $|Q|$ is invertible in $H_t(P;\mathbb{F}_p)$. An element $\alpha \in H_t(P;\mathbb{F}_p)$ is represented by $x \otimes z$, where $x$ is an element of some projective resolution $F$ of $\mathbb{Z}$ over $\mathbb{Z}P$ and $z \in \mathbb{F}_p$. Can we simply say that $|Q| \cdot (x \otimes z) = x \otimes |Q|z$ and hence $|Q|z$ has an inverse, since $\gcd(|Q|,p)=1$? Is there a canonical multiplicative structure on $H_t(P;\mathbb{F}_p)$?
I do not understand how it follows from part (1.) that $E^2_{s,t} \cong 0$ if $s > 0$.
Any help is welcome, cheers!