Сибирский федеральный университет

Тип публикации: статья из журнала

Год издания: 2021

Идентификатор DOI: 10.1134/S0081543821030196

Ключевые слова: periodic group, regular (fixed-point-free) automorphism, solvability, local finiteness, nilpotency

Аннотация: We study periodic groups of the form G = F lambda with the conditions CF (a) = 1 and vertical bar a vertical bar = 4. The mapping a : F -> F defined by the rule t -> t(a) = a(-1)ta is a fixed-pointfree (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) (Kovacs, 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) \ 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.

Журнал: PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

Выпуск журнала: Vol. 313, Is. SUPPL 1

Номера страниц: 185-193

ISSN журнала: 00815438

Место издания: NEW YORK

Издатель: MAIK NAUKA/INTERPERIODICA/SPRINGER

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

• Web of Science Core Collection