a subgroup h of a group g is called inert if‎, ‎for each $gin g$‎, ‎the index of $hcap h^g$ in $h$ is finite‎. ‎we give a classification ‎of soluble-by-finite groups $g$ in which subnormal subgroups are inert in the cases where $g$ has no nontrivial torsion normal subgroups or $g$‎ ‎is finitely generated‎.

Suppose that π be a set of primes and F a local Fitting class. Let Kπ(F) be the set of finite π-soluble groups with a Hall πsubgroup belonging to F. In this paper, we show that the class Kπ(F) is a local Fitting class. Thus, an interesting Shemetkov question for Fitting classes will be answered positively. By using the result, the F-radical of a Hall π-subgroup of a finite π-soluble group is de...

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

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 autho...

a classical result of neumann characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups. if x is a class of groups, a group g is said to have x-conjugate classes of subgroups if g/coreg(ng(h)) 2 x for each subgroup h of g. here we study groups which have soluble minimax conjugate classes of subgroups, giving a description in terms of g/z(g). w...

‎given an integer $n$‎, ‎we denote by $mathfrak b_n$ and $mathfrak c_n$ the classes of all groups $g$ for which the map $phi_{n}:gmapsto g^n$ is a monomorphism and an epimorphism of $g$‎, ‎respectively‎. ‎in this paper we give a characterization for groups in $mathfrak b_n$ and for groups in $mathfrak c_n$‎. ‎we also obtain an arithmetic description of the set of all integers $n$ such that a gr...