a Lemma of Cohomology

Aschbacher 和 Guralnick - 1984 - Some applications of the first cohomology group, Lemma 2.7(a)

Let $V$ be a $G$-module. If $N⊲G$ and $C_V(N)=0$, then $|{\mathrm{H}}^1(G,V)|⩽|{\mathrm{H}}^1(N,V)|$.

\begin{proof} Suppose $G_0$ is a complement of $V$ in $V:G$. Then $N_0:=G_0\cap (V:N)$ is a complement of $V$ in $V:N$ and $N_0⊲G_0$. Claim that $G_0=N_{V:G}(N_0)$. Otherwise, there exists a nontrivial $v\in V$ such that $N_0^v=N_0$ and so $v\in C_V(N_0)$. Since $H^0(N,V)=0$, we have $\{0\}=C_V(N)=C_V(N_0)$, which conducts a contradiction. Let $\Gamma$ and $\Gamma$ be the complements of $V$ in $V:G$ and $V:N$, respectively. Then by $G_0=N_{V:G}(N_0)$ the map

$$\begin{array}{rl}\phi :\Gamma & \to \Gamma ,\\ G_0& \mapsto N_0=G_0\cap (V:N)\end{array}$$

is injective and so $|{\mathrm{H}}^1(G,V)|⩽|{\mathrm{H}}^1(N,V)|$. \end{proof}