

Introduction




Books used




What is ``classical''?




Much used, ``Pretty'', Everyone knows it




Preserving some geometric structure:




VecSpace/GL, Orthogonal, Unitary




Some subgroups




Scalar, Diagonal, Upper triangular matrices




Diagonal Normalizer/Diagonal: Permutations




Some setup is historical quirk , some is very particular.




Classification




Related to simple groups




CFSG




Summary/History/Not proven here




The Symmetric Group




Permutations




Multiplication (from right)




cycle types, parity




Conjugacy classes: Sn=cycle type




Orbit stabilizer, class vs. centralizer




centralizer in Sn: same on cycles and permute cycles




An: Same (smaller centralizer) or split (same centralizer)




No odd perm in centralizer iff only odd cycles all different length




Ex: A5,6,7,8: 5,7,9,14 classes, Sloane: A000702




Alternating group




Proof: IsSimple (Wilson 2.3.3)




Normal subgroups of Sn, n\ge 5




Aut(An, n\ge 7) (Wilson, 2.4.1)




Wilson Lemma 2.2




Automorphism from Normalizer in Sn (Thesis, DixMort 4.2B)




Group actions




Transitive




Def. Equivalence




Action on cosets




Intransitive, embedding in direct product




Other actions of direct product




Product action




Diagonal action




Blocks, Primitivity




Definition




Correspondence Subgroups >=Stab , Blocks




Wreath Products




Abstract definition




Imprimitive Action




Sylow, Centralizer in Sn




Embedding Theorem, weak form




Product Action




When is primitive:




O'NanScott




Socle, Structure




Centralizer transitive iff group semiregular. (DM L4.2.A)




Statement (Abelian, Almost simple, Aut(T^n)




Proof DixonMortimer Thm 4.3B




Proof abelian




Otherwise isomorphic subgroup of Aut(T)\wr S_k




Diagonal




Definition




Normalizer of diagonal socle




Primitivity (DM, Thm 4.5A)




Product Action




Definition, Examples




Primitivity: DM Lemma 2.7A




Proof nonregular socle




(Cameron: Finite Permutation groups and Finite Simple Groups)




Twisted wreath




Schreier's conjecture




Degrees, Lists




Maximal Subgroups SnAn




Aschbacher's theorem




Description of Classes




Tensor products




Theorem statement




Coxeter Groups (Benson/Grove)




Reflections




Root systems




Coxeter Groups, Weyl Groups




Dynkin diagrams




Positive definite




classification




Remark about constructing in geometry




Presentations




Factor of free group




Length




Presentation




Reduction of relators




Braid groups




Algebra refresher




Fun facts for finite fields




Prime fields: mod p




Exist uniquely for p^n




p^a sub p^b iff a divides b




Field automorphisms




Chevalley Groups




Lie Algebras




How to Create




Twists




Identification with Classical Groups




Classical Groups




GL/SL




Order, subgroups




Projective Geometry




Forms




Witt's Theorem




Symplectic




Unitary




Orthogonal Char <>2




Orthogonal Char = 2




Exceptional Groups




CFSG




Sporadic Groups




Prehistory




Lie Groups


