On intersections of two non-incident subgroups of finite p-groups

Jiao Wang

doi:10.1515/math-2021-0077

Abstract

Abstract In this paper, we investigate finite p-groups G G such that whenever A , B < G A,B\lt G are non-incident, then A ∩ B ⊴ ⟨ A , B ⟩ A\cap B\hspace{0.33em}&#x22B4;\hspace{0.33em}\langle A,B\rangle . This partially solves a problem proposed by Y. Berkovich.

Highlights

All groups considered in this paper are finite
It is interesting to investigate the structure of a group G by using the intersection of two subgroups of finite p-groups
It is natural to ask the following question: What can be said about the structure of a p-group G in which H ∩ K ⊴ H and H ∩ K ⊴ K for any two non-incident subgroups H and K of G?

Summary

Introduction

All groups considered in this paper are finite. Let G be a group. It is interesting to investigate the structure of a group G by using the intersection of two subgroups of finite p-groups. Janko in [1] determined completely the structure of 2-groups G in which any two distinct maximal abelian subgroups have cyclic intersection. It is natural to ask the following question: What can be said about the structure of a p-group G in which H ∩ K ⊴ H and H ∩ K ⊴ K for any two non-incident subgroups H and K of G?. We hope to investigate the structure of a p-group G in which H ∩ K ⊴ H and H ∩ K ⊴ K for any two non-incident subgroups H and K of G (this means that H ≰ K and K ≰ H ).

Preliminaries

Some properties of -p-groups