For which pairs $(m/n)$ is every homomorphism $\mathbb Z_m->Aut(\mathbb Z_n)$ AND every homomorphism $\mathbb Z_n->Aut(\mathbb Z_m)$ trivial ?
Motivation : I want to find out, for which $m$ and $n$ does no semidirect product of $\mathbb Z_m$ and $\mathbb Z_n$ exist which is not a direct product ?
I worked out that there is only the trivial homomorphism $\mathbb Z_m->Aut(\mathbb Z_n)$ , if and only if $gcd(m,\phi(n))=1$ , where $\phi$ is the Euler-phi-function.
To get this, I used that there is only the trivial homomorphism $\mathbb Z_m->\mathbb Z_n$, if and only if $gcd(m,n)=1$
Is there an easy proof for this ?
But the problem is that I have no idea, how I can classify the pairs $(m/n)$, such that $gcd(m,\phi(n))=1$ and $gcd(n,\phi(m))=1$ both hold.
The only pair containing an even number is $(2/2)$.