The maximal subgroups of the unitary group PSU(6, q), where q=2^k

Abstract

The purpose of this paper is to study maximal subgroups of the Unitary group PSU(6, q), where q = 2^k.  The main result is a list of maximal subgroups called "the main theorem". The main theorem has been proved by using Aschbacher’s Theorem {see [1]}. Thus, this work is divided into two main parts:

Part (1): In this part, we will find the maximal subgroups in the classes C₁ – C₈ of Aschbacher’s Theorem {see [1]}.

Part (2): In this part, we will find the maximal subgroups in the class C₉ of Aschbacher’s Theorem {see [1]}, so, we will find the maximal primitive subgroups H of G which have the property that the minimal normal subgroup M of H is not abelian group and simple, thus, we divided this part into two cases:

Case (1): M is generated by transvections: In this case, we will use result of Kantor {see [9]}.

Case (2): M is a finite primitive subgroup of rank three: In this case, we will use the classification of Kantor and Liebler {see [8]}.