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ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS
Part of:
Permutation groups
Published online by Cambridge University Press: 22 October 2019
Abstract
Let 
$G$ be a finite group with two primitive permutation representations on the sets 
$\unicode[STIX]{x1D6FA}_{1}$ and 
$\unicode[STIX]{x1D6FA}_{2}$ and let 
$\unicode[STIX]{x1D70B}_{1}$ and 
$\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of 
$G$ on 
$\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of 
$G$ on 
$\unicode[STIX]{x1D6FA}_{2}$, that is, for every 
$g\in G$, 
$\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if 
$\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either 
$\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of 
$\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and 
$\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of 
$G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.
MSC classification
Primary:
20B15: Primitive groups
- Type
- Research Article
- Information
- Copyright
- © 2019 Australian Mathematical Publishing Association Inc.
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