Pypka O. Nilpotency and its generalizations in some algebraic structures.

The thesis is devoted to studying the relationships between the (generalized) central series of groups, Lie algebras, Lie rings and Leibniz algebras, and the influence of systems of subgroups on the structure of some infinite groups. Both directions are closely related to nilpotency and its generalizations in different algebraic structures. In the first case, relationships between the members and factors of finite upper and lower central series of the above structures are studied. In the second case, the influence of some types of subgroups, the presence of which as proper subgroups in nilpotent groups is impossible, is investigated. In the first part of the thesis the following results are obtained. The rank analogue of Baer's theorem for finite groups is proved. It is proved that the class of locally finite groups of finite exponent is a Baer's class. Analogues of Schur's and Baer's theorems for locally finite groups whose Sylow p-subgroups have finite exponents are proved. Similar results for locally finite groups with finite Sylow p-subgroups are obtained. It is established that the class of locally finite divisible-by-bounded groups is a Schur's class and Baer's class. A rank analogue of the generalized Baer's theorem for locally generalized radical groups is proved. The Lie analogue of the generalized Baer's theorem is proved. Analogues of Schur's and Baer's theorems for Leibniz algebras are proved. For the Lie rings we proved the analogues of Schur's and Baer's theorems. We obtained some automorphic analogues of Schur's and Baer's theorems under various natural restrictions. The second part of the thesis is devoted to the study of the influence of some natural systems of subgroups on the structure of some infinite groups. Two new generalizations of pronormal subgroups, namely GNA-subgroups and monopronormal subgroups, are constructed. We obtained the description of locally finite groups whose cyclic subgroups are GNA-subgroups (monopronormal). In particular, we proved that such groups have an abelian derived subgroup. The structure of some non-periodic groups, whose cyclic subgroups are GNA-subgroups (monopronormal) is found. More specifically, we obtained the description of non-periodic locally generalized radical groups with such properties. It is proved that in a non-periodic locally generalized radical group all subgroups are GNA-subgroups (monopronormal) if and only if it is abelian. The structure of infinite locally finite groups, whose cyclic subgroups are either ascendant or almost self-normalizing, is found. As a corollary, we obtained the description of infinite locally finite groups, whose cyclic subgroups are either subnormal or almost self-normalizing, and infinite locally finite groups, whose cyclic subgroups are either normal or almost self-normalizing.