A note on the supersolvability of a finite group with prime index of some subgroups
In this paper, we proved that a group $G$ is supersoluble if and only if for any prime $p\in π(G)$ there exists a supersoluble subgroup of index $p$.
math.GR↗
arXiv subjects
A. A. Trofimuk
Publications and source records attributed to A. A. Trofimuk.
On the supersolubility of a finite group with NS-supplemented Sylow subgroups
A subgroup $A$ of a group~$G$ is said to be {\sl NS-supplemented} in $G$, if there exists a subgroup~$B$ of $G$ such that $G=AB$ and whenever $X$~is a normal subgroup of~$A$ and $p\in π(B)$, there exists a Sylow $p$-subgroup~$B_p$ of~$B$ such that $XB_p=B_pX$. In this paper, we proved the supersolubility of a group with NS-supplemented non-cyclic Sylow subgroups. The solubility of a group with NS-supplemented maximal subgroups is obtained.
math.GR↗
On a finite group having a normal series whose factors have bicyclic Sylow subgroups
We consider the structure of a finite groups having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.
math.GR↗