A note on a class of generalized nilpotent groups introduced by Bechtell and Doerk

Abstract

All groups are finite with $\Phi (G)$ denoting the Frattini subgroup of a group G. If G is nilpotent with subgroups H and K where $H\le K,$ then $\Phi (H)\le \Phi (G)$ and $\Phi (H)\le \Phi (K).$ However, as demonstrated by the symmetric group ${\mathcal{S}}_3,$ there are non-nilpotent groups that also satisfy these two properties. In 1965 H. Bechtell introduced a class of groups that satisfy the property that $\Phi (H)\le \Phi (G)$ for all subgroups H of a group G. About 30 years later Doerk introduced a class of solvable groups that satisfy the property $\Phi (H)\le \Phi (K)$ when $H\le K\le G$ for a group G. These two classes are identical when restricted to solvable groups. In this short paper, we will extend the work done by Bechtell and Doerk by presenting some additional properties and structural results concern this class of solvable groups.

KEYWORDS AND PHRASES:

• Frattini subgroup
• nilpotent group
• Sylow p-subgroups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• PRIMARY 20F19
• Secondary: 20D20

Acknowledgement

The author is extremely grateful to the numerous helpful suggestions made by the referee that vastly improved this paper.