Connectivity of $p$-subgroup posets with irreducible characters

TL;DR

This paper studies the connectivity properties of posets formed by pairs of $p$-subgroups and their irreducible characters, revealing conditions under which these posets are disconnected and quantifying components for specific cases.

Contribution

It characterizes when the poset of $p$-subgroups with characters is disconnected and determines the number of components for certain $p$-groups, advancing understanding of subgroup-structure interactions.

Findings

$ ext{Gamma}_{p,0}(G)$ is disconnected iff } G ext{ has a strongly } p ext{-embedded subgroup or unique } p ext{-subgroup in Sylow } p ext{-subgroups}

Number of components of } ext{Gamma}_{p,1}(G) ext{ equals the order of the intersection of all } p^2 ext{-subgroups in } G

Provides new criteria for connectivity based on subgroup and character properties

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 $p^2$.