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3 Complex Representations
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abelian absolutely irreducible acter algebraic integer assumed assumption char character afforded class function CN group coherent complex valued class conjugate class coset cyclic defined degree ducible character elementary exists a normal extension field faithful character field of characteristic field of JF Frobenius group Frobenius kernel Frobenius reciprocity theorem ft(G Furthermore group with Frobenius Hence implies induction involutions irreducible character irreducible constituent isomorphic JF-representation Let 9 Let ft Let G Let H Let JF linear character matrix maximal monomial nilpotent nonabelian nonempty nonsingular normal p-complement normal subgroup p-group p'-element permutation representation prime and let prove quaternion group rational integer result follows root of unity satisfied scalar matrices Schur set of primes solvable solvable group Sp-subgroup splitting field strongly real suffices to show Suppose Sylow theorems T.I. set underlying vector space valued class function vector space