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arXiv · 2601.15266

Center-preserving irreducible representations of finite groups

Pierre-Emmanuel Caprace·Geoffrey Janssens·François Thilmany
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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.

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Pierre-Emmanuel Caprace, Geoffrey Janssens, François Thilmany. 2026-03-11. Center-preserving irreducible representations of finite groups. https://arxiv.org/abs/2601.15266

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