## On periodic groups with a regular automorphism of order 4

Тип публикации: статья из журнала

Год издания: 2019

Идентификатор DOI: 10.21538/0134-4889-2019-25-4-201-209

Ключевые слова: periodic group, regular automorphism (fixed-point-free automorphism), solvability, local finiteness, nilpotency, Local finiteness, Nilpotency, Periodic group, Regular automorphism (fixed-point-free automorphism), Solvability

Аннотация: We study periodic groups of the form G = F lambda a with the conditions C-F (a) = 1 and vertical bar a vertical bar = 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) \ 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.

#### Издание

Журнал: TRUDY INSTITUTA MATEMATIKI I MEKHANIKI URO RAN

Выпуск журнала: Vol. 25, Is. 4

Номера страниц: 201-209

ISSN журнала: 01344889

Место издания: YEKATERINBURG

Издатель: KRASOVSKII INST MATHEMATICS & MECHANICS URAL BRANCH RUSSIAN ACAD SCIENCES

#### Авторы

• Sozutov A.I (Siberian Fed Univ, Krasnoyarsk 660041, Russia)

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