On the behavior of elements of prime order from a Zinger cycle in representations of a special linear group

Let $G=SL_n(q)$, where $n\geq2$ and $q$ is a power of a prime $p$. A Zinger cycle of the group $G$ is its cyclic subgroup of order $(q^n-1)/(q-1)$. Here absolutely irreducible $G$-modules over a field of the defining characteristic $p$ where an element of a fixed prime order $m$ from a Zinger cycle of $G$ acts freely are classified in the following three cases: a) the residue of $q$ modulo $m$ generates the multiplicative group of the field of order $m$ (in particular, this holds for $m=3$); b) $m=5$; c) $n=2$.