On Sylow Subgroups of Some Shunkov Groups

Abstract

We study Shunkov groups with the following condition: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. Under this condition, we establish the structure of Sylow 2-subgroups in this group. Shunkov introduced a class of conjugate biprimitively finite groups. Since 2000, these groups are usually called Shunkov groups. In the present paper, we study Shunkov groups. A Shunkov group is defined as a group G such that, for any its finite subgroup H, any pair of conjugate elements of prime order generates a finite subgroup in the quotient group NG(H)/H. The following condition is imposed on the group: the normalizer of any nontrivial finite subgroup has an almost layer-finite periodic part. The class of groups satisfying this condition is rather wide. Thus, it contains Burnside groups of odd periods ≥ 665 [1] and Ol’shanskii groups [2]. A group is called layer-finite if the set of its elements of any given order is finite. This class was introduced by Chernikov in [3]. Almost layer-finite groups are defined as extensions of layer-finite groups with the help of finite groups. The class of almost layer-finite groups is much wider than the class of layer-finite groups. It contains all Chernikov groups. At the same time, there are Chernikov groups that are not layer-finite. Thus, in particular, this is true for the extension of a quasicyclic group with the help of reversing automorphism. If the product of all normal layer-finite subgroups of a group is layer-finite, then it is called a layer-finite radical of the group. The set of elements of finite order of a group is called a periodic part of the group if this set is a group. In Shunkov groups without almost layer-finite periodic parts such that the normalizer of any their nontrivial finite subgroup has an almost layer-finite periodic part, we study the structure of Sylow 2-subgroups. Earlier, we proved that if a Sylow 2-subgroup of the analyzed group is infinite, then it is the extension of a quasicyclic 2-group with the help of a reversing automorphism [4].