We study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii.We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group G from the normalisers of non-trivial finite subgroups of the group G if G is periodic conjugately biprimitively finite group with a strongly embedded subgroup.We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite.The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups).

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We study the structure of Sylow 2-subgroups in Shunkov periodic groups with almost layer-finite normalizers of finite nontrivial subgroups.Layer-finite groups first appeared without name in [1] and acquired their name in subsequent works Chernikov. A group is called layer-finite if the set of its elements of any given order is finite. Almost layer-finite groups are finite extensions of layer-finite groups.In the present work, we study Shunkov periodic groups (periodic conjugately biprimitively finite groups) under the condition that the normalizer of any finite nontrivial subgroup is almost layer-finite. The class of groups satisfying this condition is fairly broad. It contains free Burnside groups of odd periods ≥ 665 [2] and the groups constructed by Ol'shanskii in [3]. In such groups, we are interested in the structure of Sylow 2-subgroups.Earlier, we showed that if a Sylow 2-subgroup is infinite in groups under study, then it is a quasidihedral 2-group [4]. Recall that the extension of a quasicyclic 2-group by an inverting automorphism is called a quasidihedral 2-group (this name is explained by the fact that this group is the union of dihedral 2-groups).A group G is called a Shunkov group (a periodic conjugately biprimitively finite group) if, for any finite subgroup H of it in the quotient group N G ( H ) / H, any pair of conjugate elements of prime order generates a finite subgroup.Theorem 1. Let G be a periodic not almost layer-finite Shunkov group with finite Sylow 2-subgroup S . If the normalizer of any nontrivial finite subgroup is almost layer-finite in G, then at least one of the following assertions is true:(1) the intersection of S with the layer-finite radical of the centralizer of a central involution from S is cyclic or is a generalized group of quaternions;(2) the group S can be of one the following types: dihedral group, semidihedral group, Suzuki 2-group of order 64, Abelian group of the type ( 2 m , 2 m ) , m > 1,

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