Let $G$ be a group and $M,N$ be $G$-modules.
If $M_1$ is a $G$-submodule, then the restriction map $\operatorname{Hom}_G(M,N) \to\operatorname{Hom}_G(M_1,N)$ is injective map?
I think it is not true. But I found that someone used this argument in his proof. I am wondering whether there is some conditions on $G$-modules $M_1, M, N$ which makes this true.If such conditions exist, then it is reasonable to proceed it as he claims.
Any comments are appreciated!
Consider the exact sequence $0\to M_1\to M\to M/M_1\to 0$; applying $\operatorname{Hom}_G({-},N)$ yields the exact sequence $$ 0\to \operatorname{Hom}_G(M/M_1,N)\to \operatorname{Hom}_G(M,N)\to \operatorname{Hom}_G(M_1,N) $$ and the last map is precisely the restriction you mention.
Injectivity holds if and only if $\operatorname{Hom}_G(M/M_1,N)=\{0\}$.