Let $E$ be an elliptic curve defined over number field $K$. Let $\phi:E \to E'$ be an isogeny, and $\hat{\phi}$ be its dual isogeny.
Then, there is natural exact sequence between Selmer group, $Sel[\phi]\to Sel[2]\to Sel[\hat{\phi}]$ (Take long exact sequence Galois cohomology from short exact sequence ($ 0\to{E[\phi]}\to E[2]\to E[\hat{\phi}]$).
Why is the first map injective ? (Taking $1$-st Galois cohomology of $ 0\to{E[\phi]}\to E[2]\to E[\hat{\phi}]$, I gain $0\to{E[\phi]}\to E[2]\to E[\hat{\phi}]\to H^1(G_K,E[\phi])\to H^1(G_K,E[2])\to..$ Why is $H^1(G_K,E[\phi])\to H^1(G_K,E[2])$ injective ?)
P.S
Silverman's book, 'The arithmetic of elliptic curves', p$350$, $4$-th line from the bottom, there is exact sequence of Tate-Shafarevich group. This question is essentially the same as asking why there is the exact sequence of $p350$, $4$-th line from the bottom.