  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 $p$.
