|
This article is cited in 1 scientific paper (total in 1 paper)
AT-groups that are not AT-subgroups: Transition from $AT_{\omega}$-groups to $AT_{\Omega}$-groups
A. V. Rozhkov Kuban State University, Krasnodar
Abstract:
Periodic nonlocally finite (Burnside) groups of infinite period are studied. The first explicit example of such groups was proposed by S.V. Aleshin in 1972. His construction was generalized to AT-groups, i.e., tree automorphism groups. A number of known problems have been solved with the help of AT-groups. Up to now, in reality, only the class of $AT_{\omega}$-groups, i.e., the class of AT-groups over a sequence of cyclic groups of prime order, has been studied. In this paper, the class of $AT_{\Omega}$-groups, i.e., of AT-groups over a sequence of cyclic groups of arbitrary finite order, is studied. The difference between $AT_{\omega}$-groups and true $AT_{\Omega}$-groups was revealed by the solution of the Kourovka Problem 16.79. The study of the class of $AT_{\Omega}$-groups has allowed us to introduce a number of new notions. Now the $AT_{\omega}$-groups can be considered as elementary AT-groups by which the AT-groups over a sequence of periodic groups are saturated. We propose a strategy for studying such AT-groups and give promising directions of this kind of research.
Keywords:
Burnside groups, residually finite group, finiteness conditions, AT-groups, trees, wreath product.
Received: 09.11.2021 Revised: 18.01.2022 Accepted: 24.01.2022
Citation:
A. V. Rozhkov, “AT-groups that are not AT-subgroups: Transition from $AT_{\omega}$-groups to $AT_{\Omega}$-groups”, Trudy Inst. Mat. i Mekh. UrO RAN, 28, no. 1, 2022, 218–231
Linking options:
https://www.mathnet.ru/eng/timm1893 https://www.mathnet.ru/eng/timm/v28/i1/p218
|
Statistics & downloads: |
Abstract page: | 65 | Full-text PDF : | 12 | References: | 11 | First page: | 5 |
|