Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Gang Chen, Wenhua Zhao

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 $|I\cap Z(G)|\le |π_0(Γ_{p,e}(G))|\le {\rm Irr}(I)$, where $Γ_{p,e}(G)$, whose connected components is denoted by $π_0(Γ_{p,e}(G))$, 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.

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

Abstract

Let be a prime, a nonnegative integer, and G a finite p-group with dividing . Let I be the intersection of all subgroups of order in . It is proved that , where , whose connected components is denoted by , is the poset consisting of all pairs with , , and . Hence, an affirmative answer to Question 2 raised by Meng and Yang is obtained.
Paper Structure (3 sections, 4 theorems, 14 equations)

This paper contains 3 sections, 4 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Acknowledgment

Key Result

Lemma 2.1

(MY) Let $G$ be a finite $p$-group and $H$ a subgroup of $G$ with $|H|\ge p^{e+1}$. Then where $[(H, \varphi)]$ denotes the connected component containing $(H, \varphi)$ in $\Gamma_{p,e}(G)$.

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3