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 $|I\cap Z(G)|\le |{\pi }_0({\Gamma }_{p,e}(G))|\le \mathrm{Irr}(I)$, where ${\Gamma }_{p,e}(G)$, whose connected components is denoted by ${\pi }_0({\Gamma }_{p,e}(G))$, is the poset consisting of all pairs $(H,\phi )$ with $H\le G$, $|H|\ge p^{e+1}$, and $\phi \in \mathrm{Irr}(H)$. Hence, an affirmative answer to Question 2 raised by Meng and Yang is obtained.

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