On automorphisms of finite groups

We consider orbits of elements of a finite group $G$ with respect to the action on $G$ of a cyclic automorphism group generated by $\varphi$. We obtain sufficient conditions for the existence of an orbit whose length is equal to the order of the automorphism $\varphi$. Namely, such an orbit exists for any automorphism $\varphi$ of a semisimple or nilpotent finite group $G$ and for an automorphism $\varphi$ of an arbitrary finite group $G$ when the orders of $\varphi$ and $G$ are relatively prime. In the general case, the question of the existence of such an orbit for an automorphism of a finite group is answered negatively; a series of counterexamples is constructed. Nevertheless, the order of an automorphism $\varphi$ of a finite group $G$ is in all cases bounded by the order of $G$.
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