Arithmetic results on orbits of linear groups

Details Arithmetic results on orbits of linear groups Arithmetic results on orbits of linear groups

2012-03-12

arxiv.org/abs/1203.2457v1

Mathematics

Group Theory

Scientific paper

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well known conjecture in representation theory, and also for a longstanding question concerning 1/2-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.

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