On Periodic Groups with a Regular Automorphism of Order 4

We study periodic groups of the form $G=F\leftthreetimes\langle a\rangle$ with the conditions $C_F(a)=1$ and $|a|=4$. The mapping $a:\,F\to F$ defined by the rule $t\to t^a=a^{-1}ta$ is a fixed-point-free (regular) automorphism of the group $F$. In this case, a finite group $F$ is solvable and its commutator subgroup is nilpotent (Gorenstein and Herstein, 1961), and a locally finite group $F$ is solvable and its second commutator subgroup is contained in the center $Z(F)$ (Kovács, 1961). It is unknown whether a periodic group $F$ is always locally finite (Shumyatsky's Question 12.100 from {The Kourovka Notebook} ). We establish the following properties of groups. For $\pi=\pi(F)\setminus\pi(C_F(a^2))$, the group $F$ is $\pi$-closed and the subgroup $O_\pi(F)$ is abelian and is contained in $Z([a^2,F])$ (Theorem 1). A group $F$ without infinite elementary abelian $a^2$‑admissible subgroups is locally finite (Theorem 2). In a nonlocally finite group $F$, there is a nonlocally finite $a$-admissible subgroup factorizable by two locally finite $a$-admissible subgroups (Theorem 3). For any positive integer $n$ divisible by an odd prime, we give examples of nonlocally finite periodic groups with a regular automorphism of order $n$.