Periodic subgroups of projective linear groups in positive characteristic

Abstract

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