Let $$D$$ be a division ring and $$K$$ subfield of which is not necessarily contained in the center $$F$$ . We study structure under condition left algebraicity certain subsets over Among others, it proved that if $$D^*$$ contains noncentral normal subgroup algebraic bounded degree $$d$$ , then $$[D:F]\le d^2$$ In case $$K=F$$ obtained results show either all additive commutators or multiplicat...

The main focus in this work is to establish that L-group theory, which uses the language of functions instead formal set theoretic language, capable capturing most refined ideas and concepts classical group theory. We demonstrate by extending notion subnormality L-setting investigating its properties. develop a mechanism tackle join problem subnormal L-subgroups. conjugate L-subgroup as defined...

Abstract A finite group G G is called an MSN-group if all maximal subgroups of the Sylow are subnormal in . In this article, we investigate structure groups such that a non-MSN-group even order which every subgroup MSN-group. addition, determine minimal simple whose second MSN-groups.

We study 3-dimensional Poincaré duality pro-p groups in the spirit of work by Robert Bieri and Jonathan Hillmann, show that if such a group G has nontrivial finitely presented subnormal subgroup infinite index, then either is cyclic normal or polycyclic Demushkin an open G. Also, we describe centralizers generated subgroups groups.

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 minG. 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.