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

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 $(H, φ)$ with $H \le G$, $|H|\ge p^{e+1}$, and $φ\in {\rm Irr}(H)$. Hence, an affirmative answer to Question 2 raised by Meng and Yang is obtained.