I'm trying to find the onto, operation preserving automorphisms of ($\omega, *$), where $*$ is normal multiplication. I have only been able to think of the identity automorphism. Are there others? I know no automorphism will have the form $$f(x) = nx, n \in \omega$$ $$g(x) = x + n, n\in \omega$$ because such functions are not orperation preserving.
Moreover, if there is a 2nd automorphism, will functions similar to it be useful for constructing isomorphisms to ($\omega, *$)?
Let $P$ be the set of prime numbers. I'll leave it to you to show that if $\sigma$ is an automorphism of $(\omega,*)$, then $\sigma$ fixes $0$ and $1$ and restricts to a bijection $P\to P$.
Conversely, given any bijection $\tau\colon P\to P$, $\tau$ extends to a unique automorphism $\sigma$ by sending $0$ to $0$, $1$ to $1$, and acting on every other number according to its prime factorization: $$\sigma(p_1^{n_1}\dots p_k^{n_k})=\tau(p_1)^{n_1}\dots\tau(p_k)^{n_k}. $$
Since there are (countably) infinitely many primes, this shows that there are continuum many automorphisms of $(\omega,*)$.