Center-preserving irreducible representations of finite groups

TL;DR

The paper introduces center-preserving representations and proves that if a subgroup $H$ of a finite group $G$ has a faithful irreducible representation $\rho$, then at least one irreducible constituent of $\Ind_H^G(\rho)$ is center-preserving on $H$, yielding a new criterion related to the existence of faithful irreducibles. It develops a group-theoretic proof framework using Gaschütz's criterion, Goursat's lemma, and minimal normal subgroups, and then interprets the results in the realm of projective representations via central extensions and cohomology classes $c \in H^2(G,F^\times)$. The authors establish a detailed correspondence between center-preserving linear representations and faithful projective representations, providing criteria in terms of $\omega_\chi$ and $Z_c(G)$ that connect representation-theoretic properties with cohomological data. They also explore absolute and relative cases, present explicit examples illustrating sharpness and limitations, and discuss consequences for lifting results to Yamazaki/Schur covers, thereby enriching the understanding of the kernel structure of induced and projective representations.

Abstract

Given finite groups $H \leq G$, a representation $σ$ of $G$ is called center-preserving on $H$ if the only elements of $H$ that become central under $σ$ are those that were already central in $G$. We prove that if $H$ has a faithful irreducible representation $ρ$, then at least one of the irreducible components of the induction $\operatorname{Ind}_H^G(ρ)$ is center-preserving on $H$. In consequence, $H$ has a faithful irreducible representation if and only if every finite group $G$ containing $H$ as a subgroup has an irreducible representation whose restriction to $H$ is faithful, and which is center-preserving on $H$. In addition, we give examples illustrating the sharpness of the statement, and discuss the connection with projective representations.

Theorems & Definitions (50)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Remark 1.5
• Corollary 1.6: of \ref{['thm:faithfulcenterpreservingirrep']}
• Corollary 1.7: Corollary 5.15 in JTT25
• Theorem 2.1: See Gaschutz54, BH08
• proof
• Lemma 3.1