Periodic subgroups of projective linear groups in positive characteristic

We classify the maximal irreducible periodic subgroups of PGL(q, ), where is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and ^× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, ) containing the centre ^×1 _q of GL(q, ), such that G/ ^×1 _q is a maximal periodic subgroup of PGL(q, ), and if H is another group of this kind then H is GL(q, )-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, ) is self-normalising.