On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic

The definition of saturation condition was formulated at the end of the last century. Saturation condition has become useful in study of infinite groups. A description of various classes of infinite groups with various variants of saturating sets was obtained. In particular, it was found that periodic groups with a saturating set consisting of finite simple non-Abelian groups of Lie type, under the condition that ranks of groups in saturation set are bounded in the aggregate, are precisely locally finite groups of Lie type over a suitable locally finite field. A natural step in further research was the rejection of the periodicity condition for the group under study, and the rejection of the structure of the saturating set as a set consisting of finite simple non-Abelian groups of Lie type with ranks bounded in the aggregate. In this paper, we consider mixed Shunkov groups (i.e., groups that contain both elements of finite order and elements of infinite order).
It is well known that the Shunkov group does not have to have a periodic part (i.e., the set of elements of finite order in the Shunkov group is not necessarily a group). As a saturating set, we consider the set of full linear groups of degree 2 over finite fields of even characteristic. The lack of analogues of known results V. D. Mazurova on periodic groups with Abelian centralizers of involutions for a long time did not allow us to establish the structures of the Shunkov group with the saturation set mentioned above. In this paper, this difficulty was overcome. It is proved that a Shunkov group saturated with full linear groups of degree 2 is locally finite and isomorphic to a full linear group of degree 2 over a suitable locally finite field of characteristic 2.