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PRELIMINARIES ON ALGEBRAIC GROUPS
4 The structure of algebraic groups
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9 is defined algebraic group defined algebraic set Aut(G,T Aut(X,A Borel subgroup called character module Chevalley group classification co-torsion free connected algebraic group connected semi-simple algebraic Corollary d e r denote direct product Dynkin diagram element equivalent exists field of definition finite follows free module fundamental system G contains G defined G onto G Gal(k/k Galois GL(n GL(V group G hence homomorphism implies Inn(G inner automorphism isogeny isomorphism k-closed k-form k-isomorphism class k-rank k-rational Lemma Let G linear order maximal k-trivial torus maximal torus morphism non-trivial normal subgroup notation one-cocycle one-to-one correspondence parabolic subgroup polynomial map prime field proof Prop Proposition r-diagram of G r-fundamental Remark resp root system satisfying semi-simple algebraic groups simple groups simply connected Steinberg group Steinberg type subgroup of G subset subsystem subtorus Theorem tori torus defined torus of G trivial uniquely determined Weyl group