On groups with automorphisms generating recurrent sequences of the maximal period

Let $G$ be a finite group and $f$ be an automorphism of the group $G$. The automorphism $f$ specifies a recurrent sequence $\{ a_i \}$ on the group $G$, $i = 0, 1, \ldots$, according to the rule $a_{i+1} = f(a_i)$. If $a_0$ is the initial element of the sequence, then its period does not exceed the number of elements in the group having the same order as the element $a_0$. Thus, it makes sense to formulate the question of whether there exist groups in which such recurrent sequence for a certain automorphism has the maximal period for any initial element. In this paper we introduce the notion of an automorphism of the maximal period and find all Abelian groups and finite groups of odd orders having automorphisms of the maximal period. Also, a number of results for finite groups of even orders are established.