Let $p$ be a fixed odd prime, $K$ be a number field and $E$ be an elliptic curve defined over $K$. Set $L =K(E[p])$. We know that $G=Gal(L/K)$ is a subgroup of $GL_2(\mathbb Z/p)$.

If there is no further assumption on $G$, is it reasonable to expect that $H^1(H, E[p])$ is trivial for all subgroups $H$ of $G$?

Let $p$ be a fixed odd prime, $K$ be a number field and $E$ be an elliptic curve defined over $K$. Set $L =K(E[p])$. We know that $G=Gal(L/K)$ is a subgroup of $GL_2(\mathbb Z/p)$.

Suppose $G =GL_2(\mathbb Z/p)$, is it possible that $H^1(H, E[p])$ is trivial for all subgroups $H$ of $G$?