### 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 C_{1} – C_{8} of Aschbacher’s Theorem {see [1]}.

Part (2): In this part, we will find the maximal subgroups in the class C_{9} 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]}.**How to Cite this Article:**Rauhi I. Elkhatib, The maximal subgroups of the unitary group PSU(6, q), where q=2^k, Algebra Lett., 1 (2012), 1-21 Copyright © 2012 Rauhi I. Elkhatib. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Algebra Letters

ISSN 2051-5502

Editorial Office: office@scik.org

Copyright ©2022 SCIK Publishing Corporation