Coprime Actions and Characters Non-vanishing on the Fixed-point Subgroup
Abstract
Let a finite group A act coprimely on a finite group G and put C=CG(A). A classical theorem of Burnside asserts that the irreducible characters of a group vanishing nowhere are exactly the linear ones, and Navarro asked in Problem 21.100 of the 21st Kourovka Notebook whether the coprime analogue holds: is the number of A-invariant χ∈Irr(G) with χC nowhere zero always |C/C'|? We answer this negatively. For A cyclic of order $21$ acting on G=Brtimes V, where V=F42⊕F8 and B is the group of Boolean functions on V, the fixed subgroup C≅ C212 carries all $4096$ invariant characters while only $1728$ of them are nowhere zero on C. Because C is abelian the same example refutes Problem 6.3 of Navarro's problem list, and with it the corresponding statement about the head characters of Isaacs; passing to Grtimes A refutes Problem 6.7, a conjecture Isaacs reports is supported by abundant computational evidence. The construction needs only that V have an A-stable subset of half its size. This holds for infinitely many pairs (A,V), and for none of dimension below $7$, so the example is minimal over all operator groups of odd order. Exact machine verifications, independent of the proofs, accompany the paper.
Related
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BibTeX
@misc{hou2026coprime,
title = {{Coprime Actions and Characters Non-vanishing on the Fixed-point Subgroup}},
author = {Eric Hou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16227},
eprint = {2609.16227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16227},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coprime Actions and Characters Non-vanishing on the Fixed-point Subgroup
AU - Eric Hou
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.16227
UR - https://arxiv.org/abs/2609.16227
AB - Let a finite group A act coprimely on a finite group G and put C=C_G(A). A classical theorem of Burnside asserts that the irreducible characters of a group vanishing nowhere are exactly the linear ones, and Navarro asked in Problem 21.100 of the 21st Kourovka Notebook whether the coprime analogue holds: is the number of A-invariant χ∈Irr(G) with χ_C nowhere zero always |C/C'|? We answer this negatively. For A cyclic of order $21$ acting on G=Brtimes V, where V=F_4^2⊕F_8 and B is the group of Boolean functions on V, the fixed subgroup C≅ C_2^12 carries all $4096$ invariant characters while only $1728$ of them are nowhere zero on C. Because C is abelian the same example refutes Problem 6.3 of Navarro's problem list, and with it the corresponding statement about the head characters of Isaacs; passing to Grtimes A refutes Problem 6.7, a conjecture Isaacs reports is supported by abundant computational evidence. The construction needs only that V have an A-stable subset of half its size. This holds for infinitely many pairs (A,V), and for none of dimension below $7$, so the example is minimal over all operator groups of odd order. Exact machine verifications, independent of the proofs, accompany the paper.
ER -