Сибирский государственный университет
науки и технологий имени академика М.Ф. Решетнева

This article is about the study of infinite groups with different conditions limbs actual problem in group theory. One of such conditions is the condition of saturation of a group specified set of groups. The group is full of groups of many , if any finite subset of is contained in a subgroup isomorphic to some group of . It is known that an arbitrary periodic group, saturated groups from a variety of groups where and not fixed, is isomorphic to , where is a locally – finite field. Additionally, this result has been generalized to the case when the group is saturated with groups from a variety of groups . It would be natural to consider the case when periodic group are saturated with groups from a variety of groups : Let a periodic group is saturated with many groups , where , not fixed. Then for some locally finite fields . Thus, there is a problem of separation in periodic groups of classes of groups in which this hypothesis holds. This hypothesis is proved in the class of locally finite groups. One of the classes in which this hypothesis may be true, is the class of groups Shunkov. In this class, this hypothesis was proved for periodic groups Shunkov when the additional constraint is fixed . Attempt to abandon the fixity conditions led to the need for classification of Sylow 2-subgroups in these groups. In this work, this classification is made. The structure of Sylow 2-subgroups of the group for the case when is of the full linear group of degree two over finite fields.