Connectivity of $p$-subgroup posets with irreducible characters

Abstract

Let $G$ be a finite group. For a prime $p$ and an integer $e \geq 0$, we denote by $\Gamma_{p,e}(G)$ the set of all pairs $(H, \varphi)$, where $H$ is a $p$-subgroup of $G$ of order greater than $p^e$ and $\varphi$ is a complex irreducible character of $H$. In this paper, we investigate the connected components of the poset $\Gamma_{p,e}(G)$. For the case $e = 0$, we prove that $\Gamma_{p,0}(G)$ is disconnected if and only if either $G$ has a strongly $p$-embedded subgroup, or every Sylow $p$-subgroup of $G$ contains a unique subgroup of order $p$. Furthermore, for $e = 1$ and $G$ a $p$-group, we show that the number of connected components of $\Gamma_{p,1}(G)$ equals the order of the intersection of all subgroups of $G$ of order $p^2$.