This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in case where $D$ is non-commutative, if such exist, then either it abelian or $[D:F]<\infty$. Also, $F$ infinite field $n\geq 5$, every normal $\GL_n(F)$ abelian.

We prove that a torsion group G with all subgroups subnormal is a nilpotent group or G = N(A1×· · ·×An) is a product of a normal nilpotent subgroup N and pi -subgroups Ai , where Ai = A (i) 1 · · ·A (i) mi G , A (i) j is a Heineken–Mohamed type group, and p1, . . . , pn are pairwise distinct primes (n ≥ 1; i = 1, . . . , n; j = 1, . . . ,mi and mi are positive integers).

Abstract A subgroup X of a group G is called transitively normal if in any Y such that $$X\le Y$$ X≤Y and subnormal . Thus all subgroups are only normality transitive relation every (i.e. $$\overline{T}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:...

According to P. Hall, a subgroup $$H$$ of finite group $$G$$ is called pronormal in if, for any element $$g$$ , the subgroups and $$H^{g}$$ are conjugate $$\langle H,H^{g}\rangle$$ . The simplest examples groups normal subgroups, maximal Sylow subgroups. Pronormal were studied by number authors. For example, Legovini (1981) which every subnormal or pronormal. Later, Li Zhang (2013) described st...