A full classification of subgroups of $GL_n(q)$ containing elements of order a primitive prime divisor of $q^n-1$ is available in the literature. See here:

Robert Guralnick, Tim Penttila, Cheryl E. Praeger, and Jan Saxl. *Linear groups with orders having certain large prime
divisors*. Proc. London Math. Soc. (3), 78(1):167–214, 1999.

To give a full list of counter-examples to the OP's question, then, one needs only find which of the groups mentioned in this paper are non-abelian and lie in $SL_n(q)$.

In addition to the classical groups that Derek mentions the cited paper lists a number of `geometric' subgroups that satisfy the required conditions. These include the *field-extension subgroups* (i.e. Aschbacher's $\mathcal{C}_3$ class) whenever $n$ is a composite. So, for instance $SL_6(q)$ contains $SL_3(q^2)$ and $SL_2(q^3)$, both of which are non-abelian and have primitive prime divisors of $q^6-1$ for $q>2$.

Apart from these, all counter-examples are `nearly simple' i.e. their projective image is an almost simple group. These include the sporadic examples mentioned by Derek plus a bunch of others.

In fact the paper cited above doesn't just deal with ppd's of $q^n-1$ but ppd's of $q^e-1$ for $e>\frac{n}{2}$.

generallinear group or thespeciallinear group here? The formulation is unclear. (The motivation is also unclear, though presumably related to the tag graph-theory.) – Jim Humphreys Jul 8 '13 at 18:09