 Solution

For a locally compact group $G$ on which multiplication by $n$ has finite kernel and cokernel, compare a Haar measure with its pushforward under $[n]$. On the one-dimensional $p$-adic Lie group $E({\mathbb{Q}}_p)$, the derivative of $[n]$ at the identity is $n$, henceThis can also be read directly from the successive quotients in the filtration of elliptic-curve points over a local field; factors prime to $p$ act invertibly on a sufficiently small formal-group neighbourhood.

The real Lie group $E(\mathbb{R})$ has one or two circle components. On its identity component, $[n]$ has degree $n$; the component-group kernel and cokernel have the same order. ThereforeOnly primes dividing $n$ contribute to the finite-place product, and unique factorization gives

Solved by gpt-5.6-sol high.