Consider that a group $A$ acts by automorphism on a finite group $G$. If this action is coprime, i.e. $\gcd(|A|,|G|)=1,$ can we affirm that this action is fixed point free, i.e. $C_G(A)=1$?
I tried to think about the order of the automorphism and the order of na element of $G$, but it doesn’t work.