A refinement on a theorem of Z. Janko

TL;DR

This paper refines a theorem by Z. Janko concerning the structure of isolated subgroups within certain finite nonabelian p-groups, enhancing the understanding of subgroup isolation properties in group theory.

Contribution

It provides a refinement of Janko's theorem, offering a more precise characterization of isolated subgroups in finite nonabelian p-groups.

Findings

Refined the conditions under which subgroups are isolated in finite nonabelian p-groups

Enhanced the classification of groups with isolated subgroups

Improved understanding of subgroup intersection properties in group theory

Abstract

We say that a subgroup $H$ is isolated in a group $G$ if for each $x\in G$ we have either $x\in H$ or $\langle x\rangle \cap H={1}$. Z. Janko, in his paper [J. Algebra, 465(2016), 41--61], determined certain classes of finite nonabelian $p$-groups which possess some isolated subgroups. In this note, a theorem of his paper is refined.