I have some questions about the proof of Induction in Stages:
Induction in Stages: $\textrm{Ind}_H^G = \textrm{Ind}_N^G \textrm{Ind}_H^N$
We are fixing a representation space $(π,W)$ of $H$. <- ?? $(σ,W)$ of $H$ ?
- $T_{\sigma}(F)$ lies in $\textrm{Ind}_H^N(\sigma)$. <-?? lies in $\textrm{Ind}^G_N(σ)$ ?
$T_{\sigma}(g \cdot F)(g') = (g \cdot F)(g')(1_N) = F(g'g)(1_N) = T_{\sigma}(F)(g'g) = g \cdot T_{\sigma}(F)(g)$ <-?? $=g \cdot T_{\sigma}(F)(g′)$ ?
and don't you need a constraint $$F(gn)=π(n)F(g)$$ where $g∈G$,$n∈N$ and $(π,F(g))$ is a representation of $N$, to show $F:G→\textrm{Ind}^N_H(σ)$ lies in $\textrm{Ind}_N^G \textrm{Ind}_H^N(\sigma)$ ?