Theory and Finite Simple Groups (New Mathematical Monographs Series #8) Book

Acknowledgements     xi
List of symbols     xii
Introduction     1
Prerequisites from group theory     8
Presentations of groups     9
Generalized quaternion groups     12
2-Groups without non-cyclic abelian characteristic subgroups     15
Transfer and fusion of elements     25
Coprime group actions     34
Simple groups with dihedral or semi-dihedral Sylow 2-subgroups     38
Simple groups with strongly embedded subgroups     40
Group representations and character theory     42
Algebras, modules and representations     43
Conjugacy classes of finite groups     52
Characters of finite groups     55
Characters and algebraic integers     64
Tensor products     68
Induction and restriction     74
Frobenius groups and exceptional characters     82
Brauer's characterization of characters     90
Projective representations and central extensions     97
Modular representation theory     110
Existence of splitting p-modular systems     112
Indecomposable modules     123
Semi-perfect rings     128
Group latticesand Heller modules     137
Relative-projective modules and homomorphisms     144
Blocks of finite groups     153
Defect groups     160
Brauer's first main theorem     163
Support and kernel of a block idempotent     170
Vertices and sources     174
Modular characters of finite groups     181
Blocks of defect zero     187
Green correspondence     189
Blocks and normal subgroups     200
Blocks with normal defect groups     206
Brauer's second and third main theorems     212
Blocks of defect one     222
Group order formulas and structure theorem     242
Suzuki's group order formula     243
Thompson's group order formula     246
Groups with a unique conjugacy class of involutions     248
Brauer's group order formula     256
A theorem of G. Frobenius     259
Brauer-Suzuki Theorem     260
Glauberman's Z*-Theorem     266
Structure theorem     269
Permutation representations     277
Permutation groups     278
Orbits, stabilizers and group order     281
Conjugacy classes of permutation groups     289
Endomorphism rings of permutation modules     295
Intersection algebras     301
Character formula of Michler and Weller     307
Algorithm for computing character values     314
Completion of concrete character table calculations     316
Concrete character tables of matrix groups     320
Norton's irreducibility criterion     320
From matrix groups to permutation groups     324
Conjugacy classes of matrix groups     326
Example of a concrete character table calculation     328
Methods for constructing finite simple groups     333
Free products with an amalgamated subgroup     334
Irreducible representations of free products with amalgamated subgroups     341
Kratzer's algorithm computing compatible characters     348
Michler's algorithm constructing simple groups from given centralizers     354
Uniqueness criterion     364
Finite simple groups with proper satellites     376
Mathieu groups M[subscript 11] and M[subscript 12]     377
Mathieu groups M[subscript 22], M[subscript 23] and M[subscript 24]     388
The satellites of M[subscript 24]     397
Generating matrices of the Held group He in GL[subscript 51](11)     406
Janko's sporadic groups J[subscript 2] and J[subscript 3]     409
Simple satellites of the alternating groups     418
Janko group J[subscript 1]     433
Structure of the given centralizer     433
Character table of groups of J[subscript 1]-type     435
Existence Proof     447
Uniqueness Proof     452
Higman-Sims group HS     455
Structure of the given centralizer     455
Fusion     457
Existence proof of HS inside GL[subscript 22](11)     473
Uniqueness of HS     476
A Presentation for Aut(HS)     478
Representatives of conjugacy classes     480
Character tables     483
Harada group Ha     487
The centralizer of a 2-central involution     487
The existence proof     491
Uniqueness     497
Representatives of conjugacy classes     525
Character tables     530
Generating matrices A,B,C [isin] GL[subscript 133](19) of G     546
Thompson group Th     554
The centralizer of a 2-central involution     555
Conjugacy classes of elements of even order     557
Existence proof of Th inside GL[subscript 248](11)     561
Determination of the 3-singular conjugacy classes     571
The 5-, 7- and 13-singular conjugacy classes     598
Group order     607
Uniqueness proof and concrete character table     611
Representatives of conjugacy classes     616
Character tables     622
Partial character table of matrix group 25     641
References     645
Index     656