Groups with a strongly embedded subgroup saturated with finite simple non-abelian groups

A. A. Shlepkin

The Bulletin of Irkutsk State University. Series Mathematics, Tome 31 (2020), pp. 132-141

Résumé

An important concept in the theory of finite groups is the concept of a strongly embedded subgroup. The fundamental result on the structure of finite groups with a strongly embedded subgroup belongs to M. Suzuki. A complete classification of finite groups with a strongly embedded subgroup was obtained by G. Bender. Infinite periodic groups with a strongly embedded subgroup were first investigated by V. P. Shunkov and A. N. Izmailov under certain restrictions on the groups in question. The structure of a periodic group with a strongly embedded subgroup saturated with finite simple non-abelian groups is developed. The concepts of a strongly embedded subgroup and a group saturated with a given set of groups do not imply the periodicity of the original group. In this connection, the question arises of the location of elements of finite order both in groups with a strongly embedded subgroup and in groups saturated with some set of groups. One of the interesting classes of mixed groups (i.e., groups containing both elements of finite order and elements of infinite order) is the class of Shunkov groups. It is proved that a Shunkov group with a strongly embedded subgroup saturated with finite simple non-abelian groups has a periodic part.

Détail

Keywords: a periodic group, Shunkov group, groups saturated with a given set of groups, strongly embedded subgroup, Bender's theorem.