## Simple Groups of Lie Type |

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apply assume automorphism basis called Cartan chamber chapter characteristic Chevalley basis Chevalley group choose chosen clear coefficients combination commutator consider constants contains corresponding Coxeter complex defined definition denote describe determined dimension Dynkin diagram eigenvalues elements equivalence exists expression faces fact factorization field finite fixed follows formula fundamental roots fundamental system gallery give given Hence homomorphism identity independent induction integer invariant isomorphic joined K-character LEMMA length linear matrices multiplication N)-pair normal obtain operates orthogonal pair particular permutation polynomial positive roots PROOF properties PROPOSITION prove reflections relations rell respect result root system satisfying shown simple groups simple Lie algebra space structure subgroup subgroup of G subset Suppose symmetric THEOREM transforms twisted groups unique vector Weyl group write xr(t