On the independence of permutation characters

M. Brescia; E. Ingrosso; M. Trombetti

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arXiv · 2608.16957

On the independence of permutation characters

M. Brescia·E. Ingrosso·M. Trombetti

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Abstract

Let $G$ be a finite group. For every subgroup $H\leq G$, let $\pi_H$ be the permutation character of the action of $G$ on the left cosets of $H$. We prove that the characters $\pi_H$, with $H$ running through representatives of the conjugacy classes of subgroups of $G$, are linearly independent if and only if $G$ is cyclic. In particular, no finite insoluble group has the property asked for in Kourovka Notebook Problem~11.9. The proof uses only the fixed-point formula for a coset action and an elementary triangular-matrix argument.

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BibTeX RIS

M. Brescia, E. Ingrosso, M. Trombetti. 2026-08-16. On the independence of permutation characters. https://arxiv.org/abs/2608.16957

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