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 $C=C_G(A)$. 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 $χ\in\Irr(G)$ with ${\chi }_C$ nowhere zero always $|C/C^{\prime }|$? We answer this negatively. For $A$ cyclic of order $21$ acting on $G=B⋊V$, where $V=\F_4^2\oplus\F_8$ and $B$ is the group of Boolean functions on $V$, the fixed subgroup $C\cong C_2^{12}$ 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 $G⋊A$ 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.

23 Pages