The number of composition factors of order $p$ in completely reducible groups of characteristic $p$

Abstract

Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where $\varepsilon_q$ is a function of $q$ satisfying $1\leqslant\varepsilon_q\leqslant 3/2$. For every $q$, we give examples showing this bound is sharp infinitely often.