Notes on influence of certain permutable subgroups on finite smooth groups

Abstract

A maximal chain of a finite group $G$ is called smooth if any two intervals have the same length are isomorphic. A group $G$ is called totally smooth if all maximal chains of $G$ are smooth, and called generalized smooth if all chains from each subgroup of prime order to $G$ are smooth. In the paper entitled ``Influence of certain permutable subgroups on finite smooth groups" (A. M. Elkholy and A. A. Heliel in Acta Math. Sin. (Engl. Ser.), 27 no. 8 (2011) 1547-1556), the authors investigated the structure of finite groups which have a permutable subgroup of prime order and whose maximal subgroups are totally (or generalized) smooth groups. The results obtained by the authors require further precision. In the proof of some theorems, they overlooked some cases which may represent counterexamples to these theorems. Additionally, in certain theorems, we can omit certain hypotheses and get more accurate results. In this paper, we present counterexamples to some of these results and reintroduce these theorems after modification, using simpler and more direct proofs. Furthermore, we generalize these results by replacing certain hypotheses with weaker ones.