On Clifford-Fischer Theory Ayoub Basheer School of Mathematics, - PowerPoint PPT Presentation

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography On Clifford-Fischer Theory Ayoub Basheer ⋆ School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Pietermaritzburg Department of Mathematics, Faculty of Mathematical Sciences, University of Khartoum, P. O. Box 321, Khartoum, Sudan Jamshid Moori School of Mathematical Sciences, North-West University, Mafikeng Groups St Andrews 2013 in University of St Andrews, Scotland 3rd-11th of August 2013 Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Abstract Bernd Fischer presented a powerful and interesting technique, known as Clifford-Fischer theory , for calculating the character tables of group extensions. This technique derives its fundamentals from the Clifford theory. In this talk we describe the methods of the coset analysis and Clifford-Fischer theory applied to group extensions (split and non-split). We also mention some of the contributions to this domain and in particular of the second author and his research groups including students. Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography The Character Table of a Group Extension Let G = N · G, where N ⊳ G and G/N ∼ = G, be a finite group extension. There are several well-developed methods for calculating the character tables of group extensions. For example, the Schreier-Sims algorithm, the Todd-Coxeter coset enumeration method, the Burnside-Dixon algorithm and various other techniques. Bernd Fischer [11, 12, 13] presented a powerful and interesting technique, known nowadays as the Clifford-Fischer Theory , for calculating the character tables of group extensions. To construct the character table of G using this method, we need to have the conjugacy classes of G obtained through the coset analysis method, 1 the character tables (ordinary or projective) of the inertia factor groups, 2 the fusions of classes of the inertia factors into classes of G, 3 the Fischer matrices of G. 4 Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Coset Analysis Technique For each g ∈ G let g ∈ G map to g under the natural epimorphism π : G − → G and let g 1 = Ng 1 , g 2 = Ng 2 , · · · , g r = Ng r be representatives for the conjugacy classes of G ∼ = G/N. Therefore g i ∈ G, ∀ i, and by convention we take g 1 = 1 G . The method of the coset analysis constructs for each conjugacy class [ g i ] G , 1 ≤ i ≤ r, a number of conjugacy classes of G. For each 1 ≤ i ≤ r, we let g i 1 , g i 2 , · · · , g ic ( g i ) be the corresponding representatives of these classes. That is each conjugacy class of G corresponds uniquely to a conjugacy class of G. Also we use the notation U = π ( U ) for any subset U ⊆ G. Thus we have c ( g i ) � π − 1 ([ g i ] G ) = [ g ij ] G for any 1 ≤ i ≤ r. We assume that π ( g ij ) = g i and j =1 by convention we may take g 11 = 1 G . Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Coset Analysis Technique The coset analysis method can be described briefly in the following steps: For fixed i ∈ { 1 , 2 , · · · , r } , act N (by conjugation) on the coset Ng i and let the resulting orbits be Q i 1 , Q i 2 , · · · , Q ik i . If N is abelian (regardless to whether the extension is split or not), then | Q i 1 | = | Q i 2 | = · · · = | Q ik i | = | N | k i . Act G on Q i 1 , Q i 2 , · · · , Q ik i and suppose f ij orbits fuse together to form a new orbit ∆ ij . Let the total number of the new resulting orbits in this action be c ( g i ) (that is 1 ≤ j ≤ c ( g i ) ). Then G has a conjugacy class [ g ij ] G that contains ∆ ij and | [ g ij ] G | = | [ g i ] G | × | ∆ ij | . Repeat the above two steps, for all i ∈ { 1 , 2 , · · · , r } . Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Example of Using the Coset Analysis Technique In [10] we used the coset analysis to compute the conjugacy classes of G = 2 1+6 :((3 1+2 :8):2) . This is a maximal subgroup, of index 3, in − 2 1+6 :3 1+2 :2 S 4 , which in turn is the second largest maximal subgroup of the − − automorphism group of the unitary group U 5 (2) . Using the coset analysis we found that corresponding to the 14 classes of G = (3 1+2 :8):2 , we obtain 41 conjugacy classes for G. For example the group G has two classes of involutions represented by 2 1 and 2 2 with respective centralizer sizes 48 and 12. Corresponding to the class containing 2 2 we get five conjugacy classes in G with information listed in the following table. [ gi ] G ki mij [ gij ] G o ( gij ) | [ gij ] G | | CG ( gij ) | m 31 = 8 g 31 8 288 192 m 32 = 8 g 32 8 288 192 g 3 = 22 k 3 = 9 m 33 = 24 g 33 2 576 96 m 34 = 48 g 34 8 1728 32 m 35 = 48 g 35 4 1728 32 Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Inertia Factor Groups If G = N · G is a group extension, then G has action on the classes of N and also on Irr( N ) . Brauer Theorem (see [3] for example) asserts that the number of orbits of these two actions are the same. Let θ 1 , θ 2 , · · · , θ t be representatives of G − orbits on Irr( N ) and let H k and H k denote the corresponding inertia and inertia factor groups of θ k . In order to apply the Clifford-Fischer Theory, one have to determine the structures of all the inertia or inertia factor groups. The Clifford Theory (see [3]) deals with the character tables (ordinary or projective) of the inertia groups. Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Inertia Factor Groups In practise we do not attempt to compute the character table of H k , simply because the character tables of these inertia groups are usually much larger and more complicated to compute than the character table of G itself. Bernd Fischer suggested to use the character tables of the inertia factor groups H k together with some matrices, called by him Clifford matrices (throughout this talk we refer to them as Fischer matrices ), to construct the character table of G. Thus we firstly need to determine the structures and the appropriate projective character table of all the inertia factors H k together with the Fischer matrices. One of the biggest challenges in Clifford-Fischer theory is the determination of the type of the character table of H k (projective or ordinary), which is to be used in the construction of the character table of G. Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University

Introduction Conjugacy Classes of Group Extensions Clifford-Fischer Theory Some Survey On Clifford-Fischer Theory The Bibliography Inertia Factor Groups In practice making the right choice of the appropriate projective character table of H k , with factor set α k , might be difficult unless the Schur multipliers of all the H k are trivial. Otherwise there will be many combinations (for each H k , there are many projective character tables associated with different factor sets of the Schur multiplier of H k ) and one has to test all the possible choices and eliminate the choices that lead to contradictions. Some partial results on the extendability of characters are given in [3]. Having determined the structures and the appropriate projective character table of H k , with factor set α k (that is to be used to construct the character table of G ), the next step will be to determine the fusions of the α k − regular classes of H k into classes of G. Ayoub Basheer, Universities of KwaZulu-Natal & Khartoum Groups St Andrews 2013, St Andrews University