Connectivity of $p$-subgroup posets with irreducible characters

Abstract

Let $G$ be a finite group. For a prime $p$ and an integer $e\ge 0$, we denote by ${\Gamma }_{p,e}(G)$ the set of all pairs $(H,\phi )$, where $H$ is a $p$-subgroup of $G$ of order greater than $p^e$ and $\phi$ 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 $p^2$.

16 pages