This is a question about group representation. This is not my close area but I need to understand the following.
Let $E_1$ and $E_2$ be two elliptic curve over $\mathbb{Q}$. Let $\Lambda_1$ and $\Lambda_2$ be the $p^n$-torsion points of $E_1$ and $E_2$ respectively. Then consider the Galois groups $G_1=Gal(\mathbb{Q}(\Lambda_1)/ \mathbb{Q})$ and $G_2=Gal(\mathbb{Q}(\Lambda_2)/ \mathbb{Q})$.
How to show two Galois groups $G_1$ and $G_2$ are isomorphic to each other ?
Does that mean the corresponding Galois representations are isomorphic also ?
Which is the easiest way ?