Fusion systems on 2-groups with exactly three involutions

Abstract

Abstract We determine all saturated fusion systems ℱ on many 2-groups P with exactly three involutions. For this we use Janko's classification of these groups ([Groups of Prime Power Order. Vol. 1, Walter de Gruyter, Berlin, 2008, Section 82]) as a guide. On the way we describe all such P which contain an ℱ-essential subgroup isomorphic to C 2 k × Q 8 ${C_{2^k} \times Q_8}$ or to the Sylow 2-subgroup of PSU(3,4). Furthermore, we classify all saturated fusion systems on these P. This contributes to the outstanding classification of fusion systems on 2-groups of 2-rank 2.