Periodic subgroups of projective linear groups in positive characteristic

We classify the maximal irreducible periodic subgroups of PGL(q, F), where F is a field of positive characteristic p transcendental over its prime subfield, q ≠ p is prime, and F^× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, F) containing the centre F^× 1_q of GL(q, F), such that G/F^×1_q is a maximal periodic subgroup of PGL(q, F), and if H is another group of this kind then H is GL(q, F)-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, F) is self-normalising.