On a finite group having a normal series whose factors have bicyclic Sylow subgroups

TL;DR

This paper studies the structure of finite groups with a normal series where each factor has bicyclic Sylow subgroups, providing estimates for their derived and nilpotent lengths, especially focusing on odd order and A4-free groups.

Contribution

It offers new structural insights and precise bounds on derived and nilpotent lengths for these classes of groups, extending previous understanding.

Findings

Derived length bounds for groups with bicyclic Sylow factors

Nilpotent length estimations for specific group classes

Structural characterization of odd order and A4-free groups

Abstract

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.