Periodic subgroups of projective linear groups in positive characteristic

Alla Detinko 1  and Dane Flannery 1
  • 1 National University of Ireland

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 $$ \mathbb{F} $$ ×1q of GL(q, $$ \mathbb{F} $$ ), such that G/$$ \mathbb{F} $$ ×1q 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.

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