Connectivity of $p$-subgroup posets with irreducible characters
Hangyang Meng, Yuting Yang
TL;DR
This work studies the connectivity of the character-enhanced poset $\Gamma_{p,e}(G)$, whose elements are pairs $(H,\varphi)$ with $H$ a $p$-subgroup of $G$ of order $>p^e$ and $\varphi$ an irreducible character of $H$. A general framework reduces the connectivity problem to Sylow subgroups and Volterra-type poset maps, linking components of $\\Gamma_{p,e}(G)$ to the classical posets of $p$-subgroups. For $e=0$ the authors completely characterize disconnectivity: $\\Gamma_{p,0}(G)$ is disconnected iff $G$ has a strongly $p$-embedded subgroup or every Sylow $p$-subgroup contains a unique subgroup of order $p$; in the unique-subgroup case, the number of components is $p|G:\mathrm{N}_G(\\Omega_1(P))|$. For $p$-groups with $e=1$, they prove a sharp formula $|\\pi_0\\Gamma_{p,1}(G)|=|I|$, where $I$ is the intersection of all subgroups of order $p^2$, and provide detailed analyses of cyclic, abelian, and nonabelian order $p^3$ cases, highlighting how $I$ governs connectivity. The paper concludes with open questions about higher $e$ and classifications of $p$-groups with nontrivial $I$, tying representation theory to subgroup structure via precise combinatorial invariants.
Abstract
Let $G$ be a finite group. For a prime $p$ and an integer $e \geq 0$, we denote by $Γ_{p,e}(G)$ 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 $Γ_{p,e}(G)$. For the case $e = 0$, we prove that $Γ_{p,0}(G)$ 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 $Γ_{p,1}(G)$ equals the order of the intersection of all subgroups of $G$ of order $p^2$.