A.I. Sozutov. On periodic groups with a regular automorphism of order 4 ... P. 201-209

We study periodic groups of the form $G=F\leftthreetimes\langle a\rangle$ with the conditions $C_F(a)=1$ and $|a|=4$. 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 $Z(F)$ (Kovach, 1961). A locally finite group $F$ is solvable and its second commutator subgroup is contained in the center $Z(F)$ (Kovach, 1961). It is unknown whether a periodic group $F$ is always locally finite (Shumyatskii'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 $n$.

Keywords: periodic groups, regular automorphism (fixed-point-free automorphism), solvability, local finiteness, nilpotency

Received July 13, 2019

Revised September 30, 2019

Accepted October 21, 2019

Funding Agency: This work was supported by the Russian Foundation for Basic Research  (project № 19-01-00566 A).

Sozutov Anatoliy Ilich, Dr. Phys.-Math. Sci., Prof., Siberian Federal University, Krasnoyars, 660041 Russia, e-mail: sozutov_ai@mail.ru

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Cite this article as: A.I.Sozutov. On periodic groups with a regular automorphism of order 4, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 201–209; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 313, Suppl. 1, pp. S185–S193.