The Maximal Factorizations of the Finite Simple Groups and Their Automorphism GroupsFactorizations of finite groups as a product of two proper subgroups arise naturally in several areas of group theory, geometry, and applications. In this book, the authors determine all factorizations of the finite simple groups and their automorphism groups as a product of two maximal subgroups. The proof involved detailed study of the geometry of simple groups, and there is a substantial introductory section presenting this material. |
Contents
| 1 | |
Preliminaries | 17 |
Classical groups of large dimension factorizations as a product of two geometric subgroups | 43 |
Classical groups of large dimension factorizations in which one of the factors is nongeometric | 72 |
Classical groups of small dimension | 90 |
Factorizations of alternating groups | 123 |
Factorizations of exceptional groups of Lie type | 126 |
On maximal subgroups of symmetric and alternating groups | 130 |
Lower bounds for dimensions of representations of sporadic groups | 137 |
Uniqueness of certain representations of Suz and Cosub1 | 141 |
Orbit decomposition of Bsub4q on a spin module | 144 |
References | 146 |
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A O B absolutely irreducible acting naturally alternating group AutL B O L Chapter Chevalley groups claim that G classical groups conjugacy classes Consequently contains an element contradiction deduce Define dimension divides IAI divides IB divisible elements of order exceptional groups factorization arises field automorphism fixes G contains GF(q graph automorphism groups of Lie H of type Hence G intransitive involutions Kl-L La-Se Lemma let G let q let XA Lie type linear group Mathieu group maximal factorizations maximal in G maximal subgroups Moreover non-maximal nonsingular nonsingular 1-space nontrivial normal subgroup notation orthogonal group parabolic subgroup permutation characters primitive prime divisor Proposition 2.5 PSp2n(q q is odd q odd quadratic form representation set of 1-spaces singular 1-spaces socle sporadic groups stabilizer Subcase subgroups of G subspaces symplectic Table 4B totally singular transitive triality unitary vector whence wr S2