Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

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Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups is.ac.za ayoubbasheeriQ'ginail.coinSupervisor : Professor Jainshid Moorij amshid. I noori @ 11WU. ac.zaUNIVERSITY OF KWAZULU-NATALSchool of Mathematic

s. Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg. South AfricaA thesis submitted in the fulfillment of the requirements Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

for Philosophic Doctor (PhD) in Science at theSchool of Mathematics, Statistics and Computer Science, University ofKwaZulu-Natal, Pietermaritzburg410

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

00IN THE NAME OF ALLAH, THE MOST GRACIOUS, THE MOST MERCIFULAbstractThe character fable of a finite group is a very {Miwerful tool to study the groups

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups te simple groups, more recent work in group theory, has been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism g

roups of simple groups. The character tables of all the maximal subgroups of the sporadic simple groujis arc known, except for some maximal subgroups Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

of the Monster M and the Baby Monster B. There are several well-developed methods for calculating the character tables of group extensions and in part

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

icular when the kernel of the extension is an elementary abelian group. Character tables of finite groups can be constructed using various theoretical

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups tricen. This method derives its fundamentals from the Clifford theory. Let G — NG, where .V

cy class we construct a lion-singular square matrix 'J~i, called a Fischer matrix. Once we have all the Fischer mat rices together wit h the character Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

tables (ordinary or projective) and fusions of t he inertia fact*)!-groups into G. the character table of G is then can be constructed easily. In thi

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

s thesis wo apply the cosct analysis technique (this is a method to find the conjugacy classes of group extensions) together with theory of Clifford F

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups ons. These groups arc of the forms: 2l;+s-,1y, 37:5p(6,2), 2°s>(6:2), 2SCL(5,2), 2w:(LTs(2):2), 21+6:((31+2:8):2) and 22“'.Sp(2n, 2) and 2s'Sp(8,2). T

n addition we give some general results on the lion-split group 2»‘Sp(2n,2).iiPrefaceThe work described in this thesis was carried out under the super Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

vision and direct ion of Professor Jamshid Moon, School of Mathematics, Statistics anil Computer Sciences, University of KwaZulu-Natal. Pietermaritzbu

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

rg, from January 2009 to April 2012.The Thesis represent original work of the author and has not been otherwise been submitted in any form for any deg

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups of April 2012Date: 5th of April 2012iiiTO MY PARENTS, MY LOVELY WIFE MUNA, MY LOVELY DAUGHTER FATIMA, MY FAMILY AND TO THE BEST FRIEND I HAVE EVER GOT

MUSA COMTŨUR, I DEDICATE THIS WORK.ivAcknowledgementsFirst of all, Ĩ thank AĨ.LAĨĨ for his Gnu* and Mercy showered upon me. Ĩ heartily express niy pr Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

ofound gratitude to my supervisor. Professor Jamshid Moori, for his invaluable learned guidance, advises, encouragement, understanding and continued s

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

upport he has provided me throughout the duration of my studies which led to the compilation of•the Postgraduate Diploma project at AIMS - Cape Town 2

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups oral research. I will be always indebted lo him for introducing me to this fascinating area of Mathematics ami creat ing my interest in Group Theory.

Professor Moori is a unique encyclopaedia and I have learnt so much from him, not only in the academic orientation, but ill various walks of life. May Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

ALLAH gives him the power to enrich furthermore this interesting domain of Mathematics, ami ill general to advance the wheel of life forward.1 loving

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

ly thank my precious wife Mima. who supported me each step of the way and without her help and encouragement it simply never would have been possible

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini

Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups sity of KwaZulu-Natal. Pietermaritzburg. 1 am also grateful for the financial support that 1 have received from the University of Khartoum through the

Ministry of Higher Education of Sudan, the National Research Foundation of South Africa (NRF) for aV Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups

Clifford-Fischer Theory Applied to Certain Croups Associated with Symplcctic, Unitary and Thompson CroupsByAyoub Bashcer Mohammed Basheer ayoubac@aini