I am quite new to elliptic curves so apologies if my terminology is totally messed up.
Given an elliptic curve $E(\mathbb{F}_p):y^2 = x^3 + ax + b\text{ (mod } p)$, I am wondering if the image of multiplication map $nE = \{nP : P\in E(\mathbb{F}_p)\}$ also lies on an elliptic curve? (My intuition tells me no). If not, is there any nice way to describe the group of points?
Another question I have is I know that $E\cong \mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/m\mathbb{Z}$ for some $n, m$. For example, the curve $$E(\mathbb{F}_{23}): y^2 = x^3 + x + 2$$ is isomorphic to $\mathbb{Z}/12\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, according to pari/gp. Then the group $4E\cong\mathbb{Z}/3\mathbb{Z}$, meaning it's cyclic, verified by $$4E = \{(3, 3), (3, 20), \infty\}.$$ Recalling that for cyclic elliptic curves, there is the Additive Transfer that maps points from $E(\mathbb{F}_p)$ to $\mathbb{Z}/p\mathbb{Z}$, where we can just divide to compute discrete log efficiently. Although $4E$ is not an elliptic curve, are there similar transfers/mappings that can solve discrete log efficiently over $nE\cong\mathbb{Z}/m\mathbb{Z}$?
Thank you so much for your time.