Let $E$ be an elliptic curve defined over an algebraic number field $K$ and $X$ be a cyclic subgroup of $E(K)$ of order $p^n$, then we have the isogeny $E\rightarrow E/X$ and the dual isogeny $ E/X\rightarrow E$.

I want to ask why the kernel of the dual isogeny $ E/X\rightarrow E$ is also a cyclic group of order $p^n$. I have read most of Silverman's book(arithmetic of elliptic curves), any reference or proof is appreciated. Thanks in advance.