2000 Mathematics Subject Classification: 20F16, 20E15.Groups in which every contranormal subgroup is normally complemented has been considered. The description of such groups G with the condition Max-n and such groups having an abelian nilpotent residual satisfying Min-G have been obtained.

The current article considers some infinite groups whose finitely generated subgroups are either permutable or pronormal. A group $G$ is called a generalized radical, if $G$ has an ascending series whose factors are locally nilpotent or locally finite. The class of locally generalized radical groups is quite wide. For instance, it includes all locally finite, locally soluble, and almost locally soluble groups. The main result of this paper is the followingTheorem. Let $G$ be a locally generalized radical...

New results on tight connections among pronormal, abnormal and contranormal subgroups of a group have been established. In particular, new characteristics of pronormal and abnormal subgroups have been obtained.

We consider the unitary group U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever U acts by isometries on a metric space, every orbit is bounded. Equivalently, U is not the union of a countable chain of proper subgroups, and whenever E ⊆ U generates U, it does so by words of a fixed finite length.

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices $\mathrm{L}\left(G\right)$ and sublattices of $\mathrm{L}\left(G\right)$. It is proved that the collections of all pronormal subgroups of ${\mathrm{A}}_n$ and S${}_n$ do not form sublattices of respective $\mathrm{L}\left({\mathrm{A}}_n\right)$ and $\mathrm{L}\left({\mathrm{S}}_n\right)$, whereas the collection of all pronormal subgroups $\mathrm{LPrN}\left({\mathrm{Dic}}_n\right)$ of a dicyclic group is a sublattice of $\mathrm{L}\left({\mathrm{Dic}}_n\right)$. Furthermore, it is shown that $\mathrm{L}\left({\mathrm{Dic}}_n\right)$ and $\mathrm{LPrN}({\mathrm{Dic}}_n$) are lower semimodular lattices.