If X is a property or a class of groups, an automorphism ϕ of a group G is X-finitary if there is a normal subgroup N of G centralized by ϕ such that G/N is an X-group. Groups of such automorphisms for G a module over some ring have been very extensively studied over many years. However, for groups in general almost nothing seems to have been done. In 2009 V. V. Belyaev and D. A. Shved considered the general case for X the class of finite groups. Here we look further at the finite case but our main...

Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.

If $𝒳$ is a class of groups, then a group $G$ is said to be minimal non $𝒳$-group if all its proper subgroups are in the class $𝒳$, but $G$ itself is not an $𝒳$-group. The main result of this note is that if $c\>0$ is an integer and if $G$ is a minimal non $\left(\mathrm{ℒℱ}\right)𝒩$ (respectively, $\left(\mathrm{ℒℱ}\right){𝒩}_c$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat\left(G\right)$ is an infinite simple group; where $𝒩$ (respectively, ${𝒩}_c$, $\mathrm{ℒℱ}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally...

In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.

A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.

We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent,...

Let $G$ be a group with the property that there are no infinite descending chains of non-subnormal subgroups of $G$ for which all successive indices are infinite. The main result is that if $G$ is a locally (soluble-by-finite) group with this property then either $G$ has all subgroups subnormal or $G$ is a soluble-by-finite minimax group. This result fills a gap left in an earlier paper by the same authors on groups with the stated property.

M. V. Sapir ha formulato la seguente congettura: non esiste un semigruppo $S$ infinito, finitamente generabile, soddisfacente l'identità $x^2=0$ e immagine omomorfa di un sottosemigruppo di un gruppo $G$ nilpotente. Se ciò vale, ogni gruppo risolubile con una base finita per le sue identità semigruppali è abeliano o di esponente finito. In questo lavoro si prova la congettura di Sapir quando l'interderivato ${\gamma }_3\left(G\right)$ è periodico o se $S$ è $3$-generato e ${\gamma }_4\left(G\right)$ è periodico.

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group $G$ is finite-by-nilpotent if and only if every infinite subset contains two distinct elements $x$, $y$ such that ${\gamma }_n\left(〈x\text{,}{x}^y〉\right)$$={\gamma }_{n...}$

In this paper we deal with the class ${\mathcal{E}}_k^{*}$ of groups $G$ for which whenever we choose two infinite subsets $X$, $Y$ there exist two elements $x\in X$, $y\in Y$ such that $\left[x,\underset{k}{\underbrace{y,\mathrm{\dots },y}}\right]=1$. We prove that an infinite finitely generated soluble group in the class ${\mathcal{E}}_k^{*}$ is in the class ${\mathcal{E}}_k$ of $k$-Engel groups. Furthermore, with $k=2$, we show that if $G\in {\mathcal{E}}_2^{*}$ is infinite locally soluble or hyperabelian group then $G\in {\mathcal{E}}_2$.