The thesis examines the influence of transitively normal subgroups on the properties and structure of the group. The transitively normal subgroups form an important type of subgroups which was introduced by L.A. Kurdachenko and I. Ya. Subbotin. A subgroup H of a group G is called transitively normal in G, if H is normal in every subgroup K, such that H is subnormal in K. Note that if every subgroup of a group is transitively normal, then this group is a T-group, and conversely, every subgroup of a T-group is transitively normal. The T-groups and T -groups are quite old traditional objects of research. A wide a variety of interesting and important articles were obtained in the way of study of these subgroups. The pronormal subgroups form one of the first most studied classes of the transitively normal subgroups. Among other types of transitively normal subgroups, we can list the subgroups with the Frattini property (the weakly pronormalni subgroups), the weakly normal subgroups, the H-subgroups, the NE-subgroups, the self-conjugate permutable subgroups and others. The main new results of the dissertation are the following.1. The description of groups in which every transitively normal subgroup either normal or coincides with its normalizer were obtained for the classes of the hyperfinite groups, the FC-groups, the CC-groups, the locally finite hypofinite groups, the residually finite locally finite groups, the locally finite groups with Chernikov Sylow subgroups, the periodic almost locally nilpotent groups, the finite groups, the minimax almost soluble groups. 2. The description of the following types of groups in which every pronormal subgroup either normal or abnormal: The periodic almost locally nilpotent groups, the hyperfinite groups, the FC-groups, the locally finite hypofinite groups, the residually finite locally finite groups, the locally finite groups with Chernikov Sylow subgroups. 3. The description of finite groups, each cyclic subgroup of which is transitively normal. 4. The description of the almost locally soluble periodic groups, each infinitely generated subgroup of which is transitively normal; the locally nilpotent groups, each infinitely generated subgroup of which is transitively normal; and the radical groups, each infinitely generated subgroup of which is transitively normal.
