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.

Si studiano i gruppi risolubili non di Černikov a quozienti propri di Černikov. Nel caso periodico tali gruppi sono tutti e soli i prodotti semidiretti $H⋉N$ con $N$$p$-gruppo abeliano elementare infinito e $H$ gruppo irriducibile di automorfismi di $N$ che sia infinito e di Černikov. Nel caso non periodico invece si riconduce tale studio a quello dei moduli a quozienti...

We characterize the solvable groups without infinite properly ascending chains of non-BFC subgroups and prove that a non-BFC group with a descending chain whose factors are finite or abelian is a Cernikov group or has an infinite properly descending chain of non-BFC subgroups.

Let $G$ be a group and $n$ an integer $\ge 2$. We say that $G$ has the $n$-permutation property $(G\in P_n)$ if, for any elements $x_1,x_2,\mathrm{\cdots },x_n$ in $G$, there exists some permutation $\sigma$ of $\{1,2,\mathrm{\cdots },n\}$, $\sigma \ne id.$ such that $x_1,x_2,\mathrm{\cdots },x_n=x_{\sigma (1)},x_{\sigma (2)},\mathrm{\cdots },x_{\sigma (n)}$. We prouve that every group $G\in P_n$ is an FC-nilpotent group of class $\le n-1$, and that a finitely generated group has the $n$-permutation property (for some $n$) if, and only if, it is abelian by finite. We prouve also that a group $G\in P_3$ if, and only if, its derived subgroup has order at most 2.

Let G be a group with all subgroups subnormal. A normal subgroup N of G is said to be G-minimax if it has a finite G-invariant series whose factors are abelian and satisfy either max-G or min- G. It is proved that if the normal closure of every element of G is G-minimax then G is nilpotent and the normal closure of every element is minimax. Further results of this type are also obtained.