in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$-subnormality guarantees supersolvability‎ ‎of the whole group‎.

in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$nobreakdash-hspace{0pt}subnormality guarantees supersolvability‎ ‎of the whole group‎.

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.‎

Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...

Y. Li gave a characterization of the class of finite soluble groups in which every subnormal subgroup is normal by means of NE -subgroups: a subgroup H of a group G is called an NE -subgroup of G if NG(H) ∩ H = H. We obtain a new characterization of these groups related to the local Wielandt subgroup. We also give characterizations of the classes of finite soluble groups in which every subnorma...

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...

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.

Let $D$ be a division ring with center $F$, and $G$ subnormal or quasinormal subgroup of $D^*$. We show that if is locally solvable, then contained in $F$.