An affirmative answer to a question on connectivity of p-subgroup posets with irreducible characters

TL;DR

This paper proves a new inequality relating the structure of p-subgroup posets and irreducible characters in finite p-groups, affirmatively answering a previously posed question.

Contribution

It establishes an inequality linking the intersection of all subgroups of a certain order with the irreducible characters of a p-group, confirming a conjecture by Meng and Yang.

Findings

Proves bounds on the size of the intersection of subgroups with the set of irreducible characters.

Provides an affirmative answer to a question about the connectivity of p-subgroup posets.

Enhances understanding of the relationship between subgroup structure and character theory in finite p-groups.

Abstract

Let $p$ be a prime, $e$ a nonnegative integer, and G a finite p-group with $p^{e+1}$ dividing $|G|$. Let I be the intersection of all subgroups of order $p^{e+1}$ in $G$. It is proved that $|I\cap Z(G)|\le |\pi_0(\Gamma_{p,e}(G))|\le {\rm Irr}(I)$, where $\Gamma_{p,e}(G)$, whose connected components is denoted by $\pi_0(\Gamma_{p,e}(G))$, is the poset consisting of all pairs $(H, \varphi)$ with $H \le G$, $|H|\ge p^{e+1}$, and $\varphi\in {\rm Irr}(H)$. Hence, an affirmative answer to Question 2 raised by Meng and Yang is obtained.